This is a (forever) incomplete list of hyperbolic honeycombs. The style of this list and the names were inspired by Jonathan Bowers' list of uniform polychora, Eric's list of uniform honeycombs, and ThePokemonkey123's other list of uniform honeycombs.
Hyperbolic Tilings . . . Hyperbolic Home . . . Hyperbolic Tetracombs
Category A: Borromeachora
Category B: Prismatic Honeycombs
Category C: Antiprismatic Honeycombs
Category D: Snub Cube Borromeachora
Category E: Square Tiling Borromeachora
Category 1: Regulars - The several regular honeycombs and some of their facetings.
Category 2: Truncates - These are the truncated and quasitruncated honeycombs.
Category 3: Triangular Rectates - These are the rectates of honeycombs with triangular edge figures.
Category 4: Square Rectates - These are the rectates of honeycombs with square edge figures, along with their regiments.
Category 5: Pentagonal Rectates - These are the rectates of honeycombs with pentagonal edge figures, along with their regiments.
Category 6: Hexagonal Rectates - These are the rectates of honeycombs with hexagonal edge figures, along with their regiments.
Category 7: Sphenoverts
Category 8: Bitruncates
Category 9: Grombates
Category 10: Omnitruncates
Category 11: Prismatorhombates
Category 12: Antipodiumverts
Category 13: Podiumverts
Category 14: Square Antipodiums
Category 15: Square Podiums
Category 16: Skewverts
Category 17: Antifrustaries
Category 18: Ikhon Regiment
Category 19: Ditetrahedrals
Category 20: Cuboctahedrals
Category 21: Icosidodecahedrals
Category 22: Trihexagonals
Category 23: Miscellaneous
Category 24: Pentagonal Antipodiums
Category 25: Pentagonal Podiums
Category 26: Hexagonal Antipodiums
Category 27: Hexagonal Podiums
Category 28: Ditrigonal Prismverts
Category 29: Ditetragonal Prismverts
Category 30: Dipentagonal Prismverts
Category 31: Dihexagonal Prismverts
Category 32: Triangular Cupolaverts
Category 33: Square Cupolaverts
Category 34: Pentagonal Cupolaverts
Category 35: Hexagonal Cupolaverts
Category 36: Ditrigonal Antifrustaries
Category 37: Ditetragonal Antifrustaries
Category 38: Dipentagonal Antifrustaries
Category 39: Dihexagonal Antifrustaries
Category 40: Tetradigonaries
Category 41: Small Rhombic Tetradigonaries
Category 42: Truncate Tetradigonaries
Category 43: Great Rhombic Tetradigonaries
Category 44: Pyriticaries
Category 45: Square Tiling Ionicaries
Category 46: Triangular Tiling Ionicaries
Category 47: Snub Polyhedronaries
Category 48: x(oo^x^)/o(oo^x^) Blends
Category 49: x*(oox^)/o*(xoo^) Blends
Category 50: (x^(x^o)^o)/(x^(o^o)^x) Blends
Category 51: Phexah Compound Regiment
Category 52: Ahexah Blends
Category 53: Ashexah Blends
Category 54: Aphexah Blends