This is currently Table 6.1 of the book The Calabi problem for Fano threefolds, so more recent progress is missing at the moment.
Below the table we provide an overview of the open cases.
ID  description  moduli  $\mathrm{Aut}^0$  K‑stability  K‑polystability  K‑semistability  

11 
double cover of $\mathbb{P}^3$ with branch locus a divisor of degree 6

68 
$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
12 


$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
13 
complete intersection of quadric and cubic in $\mathbb{P}^5$ 
34 
$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
14 
complete intersection of 3 quadrics in $\mathbb{P}^6$ 
27 
$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
15 
Gushel–Mukai 3fold


$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
16 
section of halfspinor embedding of a connected component of $\mathrm{OGr}_+(5,10)$ by codimension 7 subspace 
18 
$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
17 
section of Plücker embedding of $\mathrm{Gr}(2,6)$ by codimension 5 subspace 
15 
$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
18 
section of Plücker embedding of $\mathrm{SGr}(3,6)$ by codimension 3 subspace 
12 
$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
19 
section of the adjoint $\mathrm{G}_2$Grassmannian $\mathrm{G}_2\mathrm{Gr}(2,7)$ by codimension 2 subspace 
10 
$0$ 
Kstability is not understood  general member is Kpolystable  general member is Ksemistable  
110 
zero locus of $(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$ on $\mathrm{Gr}(3,7)$ 
6 

Kstability is not understood  general member is Kpolystable but there exists one that is not  general member is Ksemistable  
111 
hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$ 
34 
$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
112 
double cover of $\mathbb{P}^3$ with branch locus a smooth quartic surface

19 
$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
113 
hypersurface of degree 3 in $\mathbb{P}^4$ 
10 
$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
114 
complete intersection of 2 quadrics in $\mathbb{P}^5$ 
3 
$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
115 
section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 3 subspace 
0 
$\mathrm{PGL}_2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
116 
hypersurface of degree 2 in $\mathbb{P}^4$ 
0 
$\mathrm{PSO}_5$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
117 
projective space $\mathbb{P}^3$ 
0 
$\mathrm{PGL}_4$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
21 
blowup of 111 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system 
36 
$0$ 
every member is Kstable  every member is Kpolystable  every member is Ksemistable  
22 
double cover of 234 with branch locus a $(2,4)$divisor 
33 
$0$ 
every member is Kstable  every member is Kpolystable  every member is Ksemistable  
23 
blowup of 112 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system 
23 
$0$ 
every member is Kstable  every member is Kpolystable  every member is Ksemistable  
24 
blowup of 117 in the intersection of two cubics

21 
$0$ 
every member is Kstable  every member is Kpolystable  every member is Ksemistable  
25 
blowup of 113 in a plane cubic 
16 
$0$ 
general member is Kstable  general member is Kpolystable  general member is Ksemistable  
26 
Verra 3fold


$0$ 
every member is Kstable  every member is Kpolystable  every member is Ksemistable  
27 
blowup of 116 in the intersection of two divisors from $\mathcal{O}_Q(2)$ 
14 
$0$ 
every member is Kstable  every member is Kpolystable  every member is Ksemistable  
28 

$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
29 
complete intersection of degree $(1,1)$ and $(2,1)$ in $\mathbb{P}^3\times\mathbb{P}^2$

13 
$0$ 
Kstability is not understood  general member is Kpolystable  general member is Ksemistable  
210 
blowup of 114 in an elliptic curve which is an intersection of 2 hyperplanes 
11 
$0$ 
Kstability is not understood  general member is Kpolystable  general member is Ksemistable  
211 
blowup of 113 in a line 
12 
$0$ 
Kstability is not understood  general member is Kpolystable  general member is Ksemistable  
212 
intersection of 3 $(1,1)$divisors in $\mathbb{P}^3\times\mathbb{P}^3$

9 
$0$ 
Kstability is not understood  general member is Kpolystable  general member is Ksemistable  
213 
blowup of 116 in a curve of degree 6 and genus 2 
8 
$0$ 
Kstability is not understood  general member is Kpolystable  general member is Ksemistable  
214 
blowup of 115 in an elliptic curve which is an intersection of 2 hyperplanes 
7 
$0$ 
Kstability is not understood  general member is Kpolystable  general member is Ksemistable  
215 

$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
216 
blowup of 114 in a conic 
7 
$0$ 
Kstability is not understood  general member is Kpolystable  general member is Ksemistable  
217 
blowup of 116 in an elliptic curve of degree 5 
5 
$0$ 
Kstability is not understood  general member is Kpolystable  general member is Ksemistable  
218 
double cover of 234 with branch locus a divisor of degree $(2,2)$ 
6 
$0$ 
every member is Kstable  every member is Kpolystable  every member is Ksemistable  
219 
blowup of 114 in a line 
5 
$0$ 
Kstability is not understood  general member is Kpolystable  general member is Ksemistable  
220 
blowup of 115 in a twisted cubic 
3 

Kstability is not understood  general member is Kpolystable but there exists one that is not  general member is Ksemistable  
221 
blowup of 116 in a twisted quartic 
2 

Kstability is not understood  general member is Kpolystable but there exists one that is not  general member is Ksemistable  
222 
blowup of 115 in a conic 
1 

general member is Kstable but there exists one that is not  general member is Kpolystable but there exists one that is not  every member is Ksemistable  
223 

$0$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
224 
divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,2)$ 
1 

Kstability is not understood  general member is Kpolystable but there exists one that is not  general member is Ksemistable  
225 
blowup of 117 in an elliptic curve which is an intersection of 2 quadrics

1 
$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
226 
blowup of 115 in a line 
0 

Kstability is not understood  none are Kpolystable  general member is Ksemistable but there exists one that is not  
227 
blowup of 117 in a twisted cubic 
0 
$\mathrm{PGL}_2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
228 
blowup of 117 in a plane cubic 
1 
$\mathbb{G}_{\mathrm{a}}^3\rtimes\mathbb{G}_{\mathrm{m}}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
229 
blowup of 116 in a conic 
0 
$\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
230 
blowup of 117 in a conic 
0 
$\mathrm{PSO}_{5;1}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
231 
blowup of 116 in a line 
0 
$\mathrm{PSO}_{5;2}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
232 
divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,1)$

0 
$\mathrm{PGL}_3$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
233 
blowup of 117 in a line 
0 
$\mathrm{PGL}_{4;2}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
234 
$\mathbb{P}^1\times\mathbb{P}^2$ 
0 
$\mathrm{PGL}_2\times\mathrm{PGL}_3$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
235 
$\mathrm{Bl}_p\mathbb{P}^3$

0 
$\mathrm{PGL}_{4;1}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
236 
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$ 
0 
$\mathrm{Aut}(\mathbb{P}(1,1,1,2))$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
31 
double cover of 327 with branch locus a divisor of degree $(2,2,2)$ 
17 
$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
32 
divisor from $\mathcal{L}^{\otimes 2}\otimes\mathcal{O}(2,3)$ on the $\mathbb{P}^2$bundle $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(1,1)^{\oplus 2})$ over $\mathbb{P}^1\times\mathbb{P}^1$ such that $X\cap Y$ is irreducible, and $\mathcal{L}$ is the tautological bundle, and $Y\in\mathcal{L}$ 
11 
$0$ 
Kstability is not understood  general member is Kpolystable  general member is Ksemistable  
33 
divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,2)$ 
9 
$0$ 
every member is Kstable  every member is Kpolystable  every member is Ksemistable  
34 
blowup of 218 in a smooth fiber of the composition of the projection to $\mathbb{P}^1\times\mathbb{P}^2$ with the projection to $\mathbb{P}^2$ of the double cover with the projection 
8 
$0$ 
every member is Kstable  every member is Kpolystable  every member is Ksemistable  
35 
blowup of 234 in a curve $C$ of degree $(5,2)$ such that $C\hookrightarrow\mathbb{P}^1\times\mathbb{P}^2\to\mathbb{P}^2$ is an embedding 
5 

general member is Kstable but there exists one that is not  general member is Kpolystable but there exists one that is not  general member is Ksemistable  
36 
blowup of 117 in the disjoint union of a line and an elliptic curve of degree 4

5 
$0$ 
general member is Kstable  general member is Kpolystable  general member is Ksemistable  
37 
blowup of 232 in an elliptic curve which is the intersection of two divisors from $\frac{1}{2}\mathrm{K}_W$ 
4 
$0$ 
Kstability is not understood  general member is Kpolystable  general member is Ksemistable  
38 
divisor from the linear system $(\alpha\circ\pi_1)^*(\mathcal{O}_{\mathbb{P}^2}(1)\otimes\pi_2^*(\mathcal{O}_{\mathbb{P}^2}(2))$ where $\pi_i\colon\mathrm{Bl}_1\mathbb{P}^2\times\mathbb{P}^2\to\mathrm{Bl}_1\mathbb{P}^2,\mathbb{P}^2$ are the projections, and $\alpha\colon\mathrm{Bl}_1\mathbb{P}^2\to\mathbb{P}^2$ is the blowup 
3 

Kstability is not understood  general member is Kpolystable but there exists one that is not  general member is Ksemistable  
39 
blowup of the cone over the Veronese of $\mathbb{P}^2$ in $\mathbb{P}^5$ with center the disjoint union of the vertex and a quartic curve on $\mathbb{P}^2$ 
6 
$\mathbb{G}_{\mathrm{m}}$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
310 
blowup of 116 in the disjoint union of 2 conics

2 

Kstability is not understood  general member is Kpolystable but there exists one that is not  every member is Ksemistable  
311 
blowup of 235 in an elliptic curve which is the intersection of two divisors from $\frac{1}{2}\mathrm{K}_{V_7}$ 
2 
$0$ 
Kstability is not understood  general member is Kpolystable  general member is Ksemistable  
312 
blowup of 117 in the disjoint union of a line and a twisted cubic 
1 

Kstability is not understood  general member is Kpolystable but there exists one that is not  every member is Ksemistable  
313 
blowup of 232 in a curve $C$ of bidegree $(2,2)$ such that the composition $C\hookrightarrow W\hookrightarrow\mathbb{P}^2\times\mathbb{P}^2\overset{p_i}{\to}\mathbb{P}^2$ is an embedding for $i=1,2$ 
1 

Kstability is not understood  general member is Kpolystable but there exists one that is not  every member is Ksemistable  
314 
blowup of 117 in the disjoint union of a plane cubic curve and a point outside the plane 
1 
$\mathbb{G}_{\mathrm{m}}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
315 
blowup of 116 in the disjoint union of a line and a conic 
0 
$\mathbb{G}_{\mathrm{m}}$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
316 
blowup of 235 in the proper transform of a twisted cubic containing the center of the blowup 
0 
$\mathrm{B}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
317 
divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$ 
0 
$\mathrm{PGL}_2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
318 
blowup of 117 in the disjoint union of a line and a conic 
0 
$\mathrm{B}\times\mathbb{G}_{\mathrm{m}}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
319 
blowup of 116 in two noncollinear points 
0 
$\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
320 
blowup of 116 in the disjoint union of two lines 
0 
$\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
321 
blowup of 234 in a curve of degree $(2,1)$ 
0 
$\mathbb{G}_{\mathrm{a}}^2\rtimes\mathbb{G}_{\mathrm{m}}^2$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
322 
blowup of 234 in a conic on $\{x\}\times\mathbb{P}^2$, $x\in\mathbb{P}^1$ 
0 
$\mathrm{B}\times\mathrm{PGL}_2$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
323 
blowup of 235 in the proper transform of a conic containing the center of the blowup 
0 
$\mathbb{G}_{\mathrm{a}}^3\rtimes(\mathrm{B}\times\mathbb{G}_{\mathrm{m}})$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
324 
the fiber product of 232 with $\mathrm{Bl}_p\mathbb{P}^2$ over $\mathbb{P}^2$

0 
$\mathrm{PGL}_{3;1}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
325 
blowup of 117 in the disjoint union of two lines

0 
$\mathrm{PGL}_{(2,2)}$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
326 
blowup of 117 in the disjoint union of a point and a line

0 
$\mathbb{G}_{\mathrm{a}}^3\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
327 
$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ 
0 
$\mathrm{PGL}_2^3$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
328 
$\mathbb{P}^1\times\mathrm{Bl}_p\mathbb{P}^2$ 
0 
$\mathrm{PGL}_2\times\mathrm{PGL}_{3;1}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
329 
blowup of 235 in a line on the exceptional divisor 
0 
$\mathrm{PGL}_{4;3,1}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
330 
blowup of 235 in the proper transform of a line containing the center of the blowup

0 
$\mathrm{PGL}_{4;2,1}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
331 
blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex

0 
$\mathrm{PSO}_{6;1}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
41 
divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of degree $(1,1,1,1)$ 
3 
$0$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
42 
blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the disjoint union of the vertex and an elliptic curve on the quadric 
2 
$\mathbb{G}_{\mathrm{m}}$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
43 
blowup of 327 in a curve of degree $(1,1,2)$ 
0 
$\mathbb{G}_{\mathrm{m}}$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
44 
blowup of 319 in the proper transform of a conic through the points 
0 
$\mathbb{G}_{\mathrm{m}}^2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
45 
blowup of 234 in the disjoint union of a curve of degree $(2,1)$ and a curve of degree $(1,0)$ 
0 
$\mathbb{G}_{\mathrm{m}}^2$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
46 
blowup of 117 in the disjoint union of 3 lines

0 
$\mathrm{PGL}_2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
47 
blowup of 232 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$ 
0 
$\mathrm{GL}_2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
48 
blowup of 327 in a curve of degree $(0,1,1)$ 
0 
$\mathrm{B}\times\mathrm{PGL}_2$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
49 
blowup of 325 in an exceptional curve of the blowup 
0 
$\mathrm{PGL}_{(2,2);1}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
410 
$\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$ 
0 
$\mathrm{PGL}_2\times\mathrm{B}^2$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
411 
blowup of 328 in $\{x\}\times E$, $x\in\mathbb{P}^1$ and $E$ the $(1)$curve 
0 
$\mathrm{B}\times\mathrm{PGL}_{3;1}$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
412 
blowup of 233 in the disjoint union of two exceptional lines of the blowup 
0 
$\mathbb{G}_{\mathrm{a}}^4\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
413 
blowup of 327 in a curve of degree $(1,1,3)$ 
1 

Kstability is not understood  general member is Kpolystable but there exists one that is not  general member is Ksemistable  
51 
blowup of 229 in the disjoint union of three exceptional lines of the blowup 
0 
$\mathbb{G}_{\mathrm{m}}$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
52 
blowup of 325 in the disjoint union of two exceptional lines on the same irreducible component 
0 
$\mathbb{G}_{\mathrm{m}}\times\mathrm{GL}_2$ 
Kstability is not understood  none are Kpolystable  none are Ksemistable  
53 
$\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$ 
0 
$\mathrm{PGL}_2\times\mathbb{G}_{\mathrm{m}}^2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
61 
$\mathbb{P}^1\times\mathrm{Bl}_4\mathbb{P}^2$ 
0 
$\mathrm{PGL}_2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
71 
$\mathbb{P}^1\times\mathrm{Bl}_5\mathbb{P}^2$ 
2 
$\mathrm{PGL}_2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
81 
$\mathbb{P}^1\times\mathrm{Bl}_6\mathbb{P}^2$ 
4 
$\mathrm{PGL}_2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
91 
$\mathbb{P}^1\times\mathrm{Bl}_7\mathbb{P}^2$ 
6 
$\mathrm{PGL}_2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable  
101 
$\mathbb{P}^1\times\mathrm{Bl}_8\mathbb{P}^2$ 
8 
$\mathrm{PGL}_2$ 
Kstability is not understood  every member is Kpolystable  every member is Ksemistable 
Open cases
Kstability is not fully understood for the following families:
 11
 12
 13
 14
 15
 16
 17
 18
 19
 110
 111
 112
 113
 114
 115
 116
 117
 28
 29
 210
 211
 212
 213
 214
 215
 216
 217
 219
 220
 221
 223
 224
 225
 226
 227
 228
 229
 230
 231
 232
 233
 234
 235
 236
 31
 32
 37
 38
 39
 310
 311
 312
 313
 314
 315
 316
 317
 318
 319
 320
 321
 322
 323
 324
 325
 326
 327
 328
 329
 330
 331
 41
 42
 43
 44
 45
 46
 47
 48
 49
 410
 411
 412
 413
 51
 52
 53
 61
 71
 81
 91
 101
Kpolystability is not fully understood for the following families:
Ksemistability is not fully understood for the following families: