Some of the information here is already given in Marek's and johannes' lists here, here, and here (and perhaps elsewhere). But the first link doesn't give vertex configurations, the second doesn't give dihedral angles, and the third only gives pentavalent vertices. So I'm making my own complete list here.
The vertex configuration (which is topological, with no regard for angles) is given first in bold. Below it are the possible vertex figures. Braces {} are for separation of face sizes and dihedral angles (and may also be interpreted as Schlafli symbols). After the verf is the solid angle, underlined; for comparison, a whole sphere is 720°. After that are the polyhedra which the verf appears in, specified by Johnson index, or Bowers acronym for a uniform polyhedron.
First let's get the odd polygons out of the way so we can focus on 3,4,5,6,8,10 later. We should certainly consider prisms up to at least n=20, because of the augmented 5,20-duoprism. (And johannes said something about n=169, but it wasn't clear.) I'm going up to n=24, and down to n=5 for some kind of completeness. The verc 5.3.3.3 only appears with the angles of an antiprism, but 4.3.3.3 and 3.3.3.3 appear with other angles, so we won't consider them yet. The vercs with n=5,6,8,10 will be listed twice: once in this post and once in the next post.
A prism's angles are obvious: 90° (or π/2 radians), and the angle of the polygon itself, (n - 2)π/n.
24.4.4
{24} 90° {4} 165° {4} 90°; 165°; 24-gonal prism
23.4.4
{23} 90° {4} 164.3478° {4} 90°; 164.3478°; 23-gonal prism
22.4.4
{22} 90° {4} 163.6364° {4} 90°; 163.6364°; 22-gonal prism
21.4.4
{21} 90° {4} 162.8571° {4} 90°; 162.8571°; 21-gonal prism
20.4.4
{20} 90° {4} 162° {4} 90°; 162°; 20-gonal prism
19.4.4
{19} 90° {4} 161.0526° {4} 90°; 161.0526°; 19-gonal prism
18.4.4
{18} 90° {4} 160° {4} 90°; 160°; 18-gonal prism
17.4.4
{17} 90° {4} 158.8235° {4} 90°; 158.8235°; 17-gonal prism
16.4.4
{16} 90° {4} 157.5° {4} 90°; 157.5°; 16-gonal prism
15.4.4
{15} 90° {4} 156° {4} 90°; 156°; 15-gonal prism
14.4.4
{14} 90° {4} 154.2857° {4} 90°; 154.2857°; 14-gonal prism
13.4.4
{13} 90° {4} 152.3077° {4} 90°; 152.3077°; 13-gonal prism
12.4.4
{12} 90° {4} 150° {4} 90°; 150°; 12-gonal prism
11.4.4
{11} 90° {4} 147.2727° {4} 90°; 147.2727°; 11-gonal prism
10.4.4
{10} 90° {4} 144° {4} 90°; 144°; dip, J20-21
9.4.4
{9} 90° {4} 140° {4} 90°; 140°; 9-gonal prism
8.4.4
{8} 90° {4} 135° {4} 90°; 135°; op, J19
7.4.4
{7} 90° {4} 128.5714° {4} 90°; 128.5714°; 7-gonal prism
6.4.4
{6} 90° {4} 120° {4} 90°; 120°; hip, J18, 54-56
5.4.4
{5} 90° {4} 108° {4} 90°; 108°; pip, J9, 52-53
An antiprism's dihedral angles are less obvious; we have the formulas
cos θn3 = (-1/√3) (sin π/n) / (1 + cos π/n),
cos θ33 = (-1/3) (4 cos π/n - 1).
So here are the antiprism verfs:
24.3.3.3
{24} 92.1687° {3} 171.3377° {3} 171.3377° {3} 92.1687°; 167.0127°; 24-gonal antiprism
23.3.3.3
{23} 92.2633° {3} 170.9609° {3} 170.9609° {3} 92.2633°; 166.4483°; 23-gonal antiprism
22.3.3.3
{22} 92.3666° {3} 170.5498° {3} 170.5498° {3} 92.3666°; 165.8327°; 22-gonal antiprism
21.3.3.3
{21} 92.4798° {3} 170.0995° {3} 170.0995° {3} 92.4798°; 165.1585°; 21-gonal antiprism
20.3.3.3
{20} 92.6043° {3} 169.6041° {3} 169.6041° {3} 92.6043°; 164.4169°; 20-gonal antiprism
19.3.3.3
{19} 92.7421° {3} 169.0566° {3} 169.0566° {3} 92.7421°; 163.5973°; 19-gonal antiprism
18.3.3.3
{18} 92.8953° {3} 168.4481° {3} 168.4481° {3} 92.8953°; 162.6868°; 18-gonal antiprism
17.3.3.3
{17} 93.0668° {3} 167.7679° {3} 167.7679° {3} 93.0668°; 161.6694°; 17-gonal antiprism
16.3.3.3
{16} 93.2598° {3} 167.0026° {3} 167.0026° {3} 93.2598°; 160.5249°; 16-gonal antiprism
15.3.3.3
{15} 93.4790° {3} 166.1351° {3} 166.1351° {3} 93.4790°; 159.2281°; 15-gonal antiprism
14.3.3.3
{14} 93.7298° {3} 165.1434° {3} 165.1434° {3} 93.7298°; 157.7464°; 14-gonal antiprism
13.3.3.3
{13} 94.0199° {3} 163.9988° {3} 163.9988° {3} 94.0199°; 156.0373°; 13-gonal antiprism
12.3.3.3
{12} 94.3592° {3} 162.6628° {3} 162.6628° {3} 94.3592°; 154.0442°; 12-gonal antiprism
11.3.3.3
{11} 94.7616° {3} 161.0833° {3} 161.0833° {3} 94.7616°; 151.6898°; 11-gonal antiprism
10.3.3.3
{10} 95.2466° {3} 159.1865° {3} 159.1865° {3} 95.2466°; 148.8663°; dap, J24-25
9.3.3.3
{9} 95.8430° {3} 156.8662° {3} 156.8662° {3} 95.8430°; 145.4184°; 9-gonal antiprism
8.3.3.3
{8} 96.5945° {3} 153.9624° {3} 153.9624° {3} 96.5945°; 141.1138°; oap, J23
7.3.3.3
{7} 97.5723° {3} 150.2223° {3} 150.2223° {3} 97.5723°; 135.5890°; 7-gonal antiprism
6.3.3.3
{6} 98.8994° {3} 145.2219° {3} 145.2219° {3} 98.8994°; 128.2426°; hap, J22
5.3.3.3
{5} 100.8123° {3} 138.1897° {3} 138.1897° {3} 100.8123°; 118.0040°; pap, J11, 62-64, 92