There are 15 (para)compact honeycombs, and a few starry noncompact honeycombs are listed here as well. There are also some facetings of the honeycombs that are tho analogues (aka the freeloaders). All honeycombs here are tame.
These are the compact honeycombs, which means they have finite cells and verfs. All are orientable.
1. Ikhon - (ike HON) icosahedral honeycomb. It has ikes for cells. Its verf is a doe. I call its symmetry group "ikhonnic". It is the most ikhonic hyperbolic honeycomb.
2. Doehon - (DOE hon) dodecahedral honeycomb. It has does for cells. Its verf is an oct. I call its symmetry group "doehonnic".
3. Pechon - (PECH hon) order-5 cubic honeycomb. It has cubes for cells. Its verf is an ike. It has doehonnic symmetry.
4. Pedhon - (PED hon) order-5 dodecahedral honeycomb. It has does for cells. Its verf is an ike. I call its symmetry "pedhonnic".
These have infinitely large cells or verfs. All are orientable.
5. Hexah - (hex AH) hexagonal tiling honeycomb. It has hexats for cells. Its verf is a tet. I call its symmetry "hexahic"
6. Thon - (THON) tetrahedral honeycomb. It has tets for cells. Its verf is a trat. It has hexahic symmetry.
7. Shexah - (shex AH) order-4 hexagonal tiling honeycomb. It has hexats for cells. Its verf is an oct. Its symmetry is called "shexahic"
8. Hachon - (HA chon) order-6 cubic honeycomb. It has cubes for cells. Its verf is a trat. It has shexahic symmetry.
9. Phexah - (fex AH) order-5 hexagonal tiling honeycomb. It has hexats for cells. Its verf is an ike. Its symmetry is called "phexahic".
10. Hedhon - (HED hon) order-6 dodecahedral honeycomb. It has does for cells. Its verf is a trat. It has phexahic symmetry.
11. Hihexah - (HI hex ah)order-6 hexagonal tiling honeycomb. It has hexats for cells. Its verf is a trat. Its symmetry is called "hihexahic".
12. Trah - (TRA) triangular tiling honeycomb. It has trats for cells. Its verf is a hexat. Its symmetry is called "trahic".
13. Squah - (SQUA) square tiling honeycomb. It has squats for cells. Its verf is a cube. It has "squahic" symmetry.
14. Octh - (OCTH) octahedral honeycomb. It has octs for cells. Its verf is a squat. It has squahic symmetry.
15. Sisquah - (SIS qua) order-4 square tiling honeycomb. It has squats for cells. Its verf is a squat. It has sisquahic symmetry.
These have hyperbolic cells or verfs. All are orientable.
16. Fipech - (fi PECH) faceted pechon. It has pesquats for cells. Its verf is a gad. It has doehonnic symmetry.
17. Siddoh - (SID oh) small stellated doehon. It has sissids for cells. Its verf is a peat. It has doehonnic symmetry.
18. Fipped - (fi PED) faceted pedhon. It has pepats for cells. Its verf is a gad. It has pedhonnic symmetry.
19. Sipped - (SIP ped) small stellated pedhon. It has sissids for cells. Its verf is a pepat. It has pedhonnic symmetry.
20. Fiphexah - (fi fex AH) faceted phexah. It has phexats for cells. Its verf is a gad. It has hedhonnic symmetry.
21. Shad - (SHAD) small stellated hedhon. It has sissids for cells. Its verf is a hipat. It has hedhonnic symmetry.
These are tho analogues, and have either thah, ditatha, or sha verfs. Dihipesh and hihish are nonorientable.
22. Dihipesh - (DIH hi pesh) doe-hemipesquat honeycomb. It has does and peats.
23. Dithah - (DITH ah) ditrigonal tet-hemiazat honeycomb. It has tets and aztrats.
24. Hihish - (HI hish) hexat-hemihisquat honeycomb. It has hexats and shexats.
25. Ditchissah - (dit CHISS ah) ditrigonal cube-hemisquazat honeycomb. It has cubes and asquats.
26. Ditdohpah - (DIT doep ah)ditrigonal doe-hemipazat honeycomb. It has does and azpats.
27. Dithehihah - (DITH eh hi ha) ditrigonal hexat-hemihazat honeycomb. It has hexats and azhexats.
28. Ohah - (OH ha) oct-hemiazat honeycomb. It has octs and aztrats.
29. Shahsah - (SHA sa) squat-hemisquazat honeycomb. It has squats and asquats.