Enumerate subsets of edges of a cube that are distinct over rotations. (I've seen a photo of a sculpture of this.) Other additional constraints to consider:
Stable on all faces: every vertex must participate in at least on edge. This is a stronger criterion than the center of mass being over the face in contact with the ground.
Stable on at least one face. Then consider only rotations around the axis that goes through stable faces.
Add face and space diagonals and allow the constraint of rigidness: every joint is a ball and socket joint. Maybe if diagonals intersect, assume a joint there, too.
Other graphs, for example the wireframe of 8 cubes forming a larger cube.