In his Mathematical Games column for September 1958, Martin Gardner discussed the Soma cube, a puzzle which was later marketed in North America by Parker Brothers.
Later, in September 1972, his column in Scientific American discussed polycube puzzles in general. In addition to the Soma cube, it discussed the Diabolical Cube, which dated from the Victorian era, and the cube devised by J. G. Mikusiński, a Polish mathematician.
A later addendum to the article mentioned another cube devised by Lesk Kokay, first brought to light in New Zealand.
These cube puzzles are depicted below:
Except for the Soma cube, though, these puzzles are not readily available. However, I have seen what I thought was the Mikusiński cube in an advertisement, and I may well have missed its commercial availability.
One set of cube puzzles that was readily available for a time - and which seems to have returned to the market - are the Impuzzables, originally sold by Leisure Dynamics. Apparently, they are currently offered by a company named Puzzle Master.
The design of the Impuzzables has been variously credited to either Gerard d'Arcey, an independent game inventor working in California, or to Robert Beck, working at Custom Concepts.
While the Perfect White cube is very similar to a mirror reflection of the Mikusiński cube, it differs from that in that the second and third pieces are identical instead of mirror reflections of each other.
In fact, this page (which is a descendant of this page, not having backward links on it) notes that of twelve possible puzzles that can be formed from the Mikusiński cube by mirror reflections of one or more pieces, only one of them has no solution in the form of a 3 by 3 by 3 cube, and all the rest have only a small number of possible solutions, thus presumably making them difficult puzzles.
The following pages contain spoilers, giving the solutions to many of the cube puzzles discussed here: