The big UFO on the main menu initially contains seven rockets full of puzzles, but more ships with extra puzzles can be unlocked by completing them.
In each of the four story mode chapters, there are 5 special puzzles that can unlock items for your character. In order to find them, you have to complete certain puzzles quickly. It might take multiple playthroughs of the chapter. Once you've found all 5 and completed the chapter, a special puzzle will unlock for that chapter. To access it, click on the orange rocket, and then the blue alien ship on the lowest row.