Hello, puzzlers!
After a lot of work and deciding, I have come up with a complete? table of 3x3 mods and their combinations.
Notice the ?. If I am forgetting any, PLEASE mention.
First, I will say what I am not including. If I am not including something, that means that either it is too rare/ not a *common* mod, or it is something with a TON of possibilities (AKA sticker mods).
So here is what will not be in the list:
Sticker mods (this includes turning a slice, and then restickering)
Siamese/ fused
Unique/ irregular/ not common
Mini/ large
Gears
Mixup
Bandaged
Hybrids
Void/ brilicube
Weird extensions
Circle/ crazy
Restricted turning (slice turn only, 180 degrees on one axis, etc.)
Babyface
Rhombic Dodecahedron
Borg
Wall cube types (different layers have different types of pieces. Ex. viewtopic.php?f=15&t=23952)
Bicone
Barrel

Now here is the list of possible mods.
Normal 3x3
Fisher
Windmill
Cuboctahedron
Hexagonal Dipyramid
Axis (standard)
Half Truncated
Tetrahedron
Octahedron
Ghost
Sphere

If all of them are considered equal, and you can have 2 mods per cube, that means there are 121 possible combinations.
However, they are not all equal. For example, if they were all equal, then you could have a Tetrahedron Cuboctahedron. This is not possible.
So I have divided them up into 2 types- shape, and pieces.
So here is that list again, divided into sub catagories (Note that I have divided normal 3x3 into 3x3 and Hexahedron)

Shape:
Hexahedron
Cuboctahedron
Hexagonal Dipyramid
Half Truncated
Tetrahedron
Octahedron
Sphere

Piece:
Normal 3x3
Fisher
Windmill
Axis (Standard)
Ghost

Now, you can only choose one from each category. That means there is a total of 35 possible combinations.
I will now list them all. The format will be SHAPE - PIECE. If a combination has a common other name, it will look like this SHAPE - PIECE (NAME)

Hexahedron - Normal 3x3 (3x3 Rubik's cube)
Hexahedron - Fisher (Fisher cube)
Hexahedron - Windmill (Windmill cube)
Hexahedron - Axis (Axis cube)
Hexahedron - Ghost (Ghost cube)

Cuboctahedron - Normal 3x3 (Cuboctahedron)
Cuboctahedron - Fisher
Cuboctahedron - Windmill
Cuboctahedron - Axis
Cuboctahedron - Ghost

Hexagonal Dipyramid - Normal 3x3 (Hexagonal Dipyramid)
Hexagonal Dipyramid - Fisher
Hexagonal Dipyramid - Windmill
Hexagonal Dipyramid - Axis
Hexagonal Dipyramid - Ghost

Half Truncated - Normal 3x3 (Half Truncated cube)
Half Truncated - Fisher
Half Truncated - Windmill
Half Truncated - Axis
Half Truncated - Ghost

Tetrahedron - Normal 3x3 (Master Morphinx)
Tetrahedron - Fisher
Tetrahedron - Windmill
Tetrahedron - Axis
Tetrahedron - Ghost

Octahedron - Normal 3x3 (Trajbers Octahedron)
Octahedron - Fisher
Octahedron - Windmill
Octahedron - Axis
Octahedron - Ghost

Sphere - Normal 3x3 (Ball cube)
Sphere - Fisher
Sphere - Windmill
Sphere - Axis
Sphere - Ghost

Am I missing any shapes or pieces?
Thanks!
-Doug

Why did you select this shapes?

You can take every shape feom the Platon solids, or Archimedean or Catalan, or Kepler and much more...

Andreas built following 3x3 mods: PentagonalDodec, Icosahedron, Rhombtriacontahedron, Triakis .... (cuts only, no extensions)

To build the Tetrahedron you need extensions. In thsis case you must allow ALL shapes of the categories I listed.

The HD is not include above. It is a very nice looking shape. What is with the Cheops Pyramide, or Trajbers ...?

I only included the common place shapes. Andreas's mods are interesting, but they are rare.

The HD is there, belive me And what do you mean, what's with the Trajbers octahedron?

I for myself think the rhombic dodecahedron shouldn't be left out. There are currently 33 puzzles with this shape in the museum.
There has been a time when the hexagonal dipyramid was rare.

Sorry, but how about my Mirror Blocks?
It is mass-produced, and common 3x3 type, and hexahedron.

I don't know of any mirror blocks mods besides gluing things onto it and bumping other cubes (2x2, pyraminx, etc)

-Doug

Hidetoshi is saying that his Mirror Blocks can be added to your list of Hexahedron puzzles. I think there are a couple of Mirror Blocks mods too. I'm sure Traiphum Created a Bump Megamorphinx, and I think there is a Hexagonal Dipyramid Mirror Blocks too along with some others.