This table tries to give the state of the art of the ongoing project of understanding Kstability for all Fano 3folds. Let me know if I have made a mistake, or am missing some recent progress!
Below the table we provide an overview of the open cases.
ID  description  moduli  $\mathrm{Aut}^0$  K‑stability  K‑polystability  K‑semistability  references  

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

68 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
12 


$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
13 
complete intersection of quadric and cubic in $\mathbb{P}^5$ 
34 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
14 
complete intersection of 3 quadrics in $\mathbb{P}^6$ 
27 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
15 
Gushel–Mukai 3fold


$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
16 
section of halfspinor embedding of a connected component of $\mathrm{OGr}_+(5,10)$ by codimension 7 subspace 
18 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
17 
section of Plücker embedding of $\mathrm{Gr}(2,6)$ by codimension 5 subspace 
15 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
18 
section of Plücker embedding of $\mathrm{SGr}(3,6)$ by codimension 3 subspace 
12 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
19 
section of the adjoint $\mathrm{G}_2$Grassmannian $\mathrm{G}_2\mathrm{Gr}(2,7)$ by codimension 2 subspace 
10 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
110 
zero locus of $(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$ on $\mathrm{Gr}(3,7)$ 
6 

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

19 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
113 
hypersurface of degree 3 in $\mathbb{P}^4$ 
10 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
114 
complete intersection of 2 quadrics in $\mathbb{P}^5$ 
3 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
115 
section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 3 subspace 
0 
$\mathrm{PGL}_2$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
116 
hypersurface of degree 2 in $\mathbb{P}^4$ 
0 
$\mathrm{PSO}_5$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
117 
projective space $\mathbb{P}^3$ 
0 
$\mathrm{PGL}_4$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
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 K‑stable  every member is K‑polystable  every member is K‑semistable  
22 
double cover of 234 with branch locus a $(2,4)$divisor 
33 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
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 K‑stable  every member is K‑polystable  every member is K‑semistable  
24 
blowup of 117 in the intersection of two cubics

21 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
25 
blowup of 113 in a plane cubic 
16 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
26 
Verra 3fold


$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
27 
blowup of 116 in the intersection of two divisors from $\mathcal{O}_Q(2)$ 
14 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
28 

$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
29 
complete intersection of degree $(1,1)$ and $(2,1)$ in $\mathbb{P}^3\times\mathbb{P}^2$

13 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
210 
blowup of 114 in an elliptic curve which is an intersection of 2 hyperplanes 
11 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
211 
blowup of 113 in a line 
12 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
212 
intersection of 3 $(1,1)$divisors in $\mathbb{P}^3\times\mathbb{P}^3$

9 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
213 
blowup of 116 in a curve of degree 6 and genus 2 
8 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
214 
blowup of 115 in an elliptic curve which is an intersection of 2 hyperplanes 
7 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
215 

$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
216 
blowup of 114 in a conic 
7 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
217 
blowup of 116 in an elliptic curve of degree 5 
5 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
218 
double cover of 234 with branch locus a divisor of degree $(2,2)$ 
6 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
219 
blowup of 114 in a line 
5 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
220 
blowup of 115 in a twisted cubic 
3 

general member is K‑stable but there exists one that is not  general member is K‑polystable but there exists one that is not  general member is K‑semistable  
221 
blowup of 116 in a twisted quartic 
2 

general member is K‑stable but there exists one that is not  general member is K‑polystable but there exists one that is not  general member is K‑semistable  
222 
blowup of 115 in a conic 
1 

general member is K‑stable but there exists one that is not  general member is K‑polystable but there exists one that is not  every member is K‑semistable  
223 

$0$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
224 
divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,2)$ 
1 

general member is K‑stable but there exists one that is not  general member is K‑polystable but there exists one that is not  general member is K‑semistable  
225 
blowup of 117 in an elliptic curve which is an intersection of 2 quadrics

1 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
226 
blowup of 115 in a line 
0 

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

0 
$\mathrm{PGL}_3$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
233 
blowup of 117 in a line 
0 
$\mathrm{PGL}_{4;2}$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
234 
$\mathbb{P}^1\times\mathbb{P}^2$ 
0 
$\mathrm{PGL}_2\times\mathrm{PGL}_3$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
235 
$\mathrm{Bl}_p\mathbb{P}^3$

0 
$\mathrm{PGL}_{4;1}$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
236 
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$ 
0 
$\mathrm{Aut}(\mathbb{P}(1,1,1,2))$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
31 
double cover of 327 with branch locus a divisor of degree $(2,2,2)$ 
17 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
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 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
33 
divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,2)$ 
9 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
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 K‑stable  every member is K‑polystable  every member is K‑semistable  
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 K‑stable but there exists one that is not  general member is K‑polystable but there exists one that is not  general member is K‑semistable  
36 
blowup of 117 in the disjoint union of a line and an elliptic curve of degree 4

5 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
37 
blowup of 232 in an elliptic curve which is the intersection of two divisors from $\frac{1}{2}\mathrm{K}_W$ 
4 
$0$ 
general member is K‑stable  general member is K‑polystable  general member is K‑semistable  
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 

general member is K‑stable but there exists one that is not  general member is K‑polystable but there exists one that is not  general member is K‑semistable  
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 
$\mathbb{G}_{\mathrm{m}}$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
310 
blowup of 116 in the disjoint union of 2 conics

2 

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

general member is K‑stable but there exists one that is not  general member is K‑polystable but there exists one that is not  every member is K‑semistable  
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 

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

0 
$\mathrm{PGL}_{3;1}$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
325 
blowup of 117 in the disjoint union of two lines

0 
$\mathrm{PGL}_{(2,2)}$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
326 
blowup of 117 in the disjoint union of a point and a line

0 
$\mathbb{G}_{\mathrm{a}}^3\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
327 
$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ 
0 
$\mathrm{PGL}_2^3$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
328 
$\mathbb{P}^1\times\mathrm{Bl}_p\mathbb{P}^2$ 
0 
$\mathrm{PGL}_2\times\mathrm{PGL}_{3;1}$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
329 
blowup of 235 in a line on the exceptional divisor 
0 
$\mathrm{PGL}_{4;3,1}$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
330 
blowup of 235 in the proper transform of a line containing the center of the blowup

0 
$\mathrm{PGL}_{4;2,1}$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
331 
blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex

0 
$\mathrm{PSO}_{6;1}$ 
none are Kstable  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 
$0$ 
every member is K‑stable  every member is K‑polystable  every member is K‑semistable  
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 
$\mathbb{G}_{\mathrm{m}}$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
43 
blowup of 327 in a curve of degree $(1,1,2)$ 
0 
$\mathbb{G}_{\mathrm{m}}$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
44 
blowup of 319 in the proper transform of a conic through the points 
0 
$\mathbb{G}_{\mathrm{m}}^2$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
45 
blowup of 234 in the disjoint union of a curve of degree $(2,1)$ and a curve of degree $(1,0)$ 
0 
$\mathbb{G}_{\mathrm{m}}^2$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
46 
blowup of 117 in the disjoint union of 3 lines

0 
$\mathrm{PGL}_2$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
47 
blowup of 232 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$ 
0 
$\mathrm{GL}_2$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
48 
blowup of 327 in a curve of degree $(0,1,1)$ 
0 
$\mathrm{B}\times\mathrm{PGL}_2$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
49 
blowup of 325 in an exceptional curve of the blowup 
0 
$\mathrm{PGL}_{(2,2);1}$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
410 
$\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$ 
0 
$\mathrm{PGL}_2\times\mathrm{B}^2$ 
none are Kstable  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 
$\mathrm{B}\times\mathrm{PGL}_{3;1}$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
412 
blowup of 233 in the disjoint union of two exceptional lines of the blowup 
0 
$\mathbb{G}_{\mathrm{a}}^4\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
413 
blowup of 327 in a curve of degree $(1,1,3)$ 
1 

general member is K‑stable but there exists one that is not  general member is K‑polystable but there exists one that is not  general member is K‑semistable  
51 
blowup of 229 in the disjoint union of three exceptional lines of the blowup 
0 
$\mathbb{G}_{\mathrm{m}}$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
52 
blowup of 325 in the disjoint union of two exceptional lines on the same irreducible component 
0 
$\mathbb{G}_{\mathrm{m}}\times\mathrm{GL}_2$ 
none are Kstable  none are Kpolystable  none are Ksemistable  
53 
$\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$ 
0 
$\mathrm{PGL}_2\times\mathbb{G}_{\mathrm{m}}^2$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
61 
$\mathbb{P}^1\times\mathrm{Bl}_4\mathbb{P}^2$ 
0 
$\mathrm{PGL}_2$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
71 
$\mathbb{P}^1\times\mathrm{Bl}_5\mathbb{P}^2$ 
2 
$\mathrm{PGL}_2$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
81 
$\mathbb{P}^1\times\mathrm{Bl}_6\mathbb{P}^2$ 
4 
$\mathrm{PGL}_2$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
91 
$\mathbb{P}^1\times\mathrm{Bl}_7\mathbb{P}^2$ 
6 
$\mathrm{PGL}_2$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable  
101 
$\mathbb{P}^1\times\mathrm{Bl}_8\mathbb{P}^2$ 
8 
$\mathrm{PGL}_2$ 
none are Kstable  every member is K‑polystable  every member is K‑semistable 
Open cases
Kstability is not fully understood for the following families:
Kpolystability is not fully understood for the following families:
Ksemistability is not fully understood for the following families: