After admitting a few pillows with straight edges, there is no end to it. Here are all 24 pillows based on a square that either have a straight, concave, or convex edge. We disregard rotational copies but keep mirrors.
Usually, polyformists try to tile simple shapes using each polyform exactly once. The archetypical example is to tile a 6×10 rectangle with all 12 pentominoes. This is in most cases a tedious exercise that doesn’t teach you much more than backtracking. On the other hand, nothing is worse than not knowing, so here you go: Three puzzles that ask to tile the outlined region by using each of the 24 pillows exactly once.
The grid is there to help placing the pillows. These puzzles are actually not so bad. The first one for instance requires to make economic use of the pillows with straight edges. I post the solutions below, mainly because nobody would do them anyway and to prevent future waste of time.
Note in the solution above the second column consisting entirely of pillows with parallel straight edges. I think this has to appear in any solution of this puzzle.
The one above is my favorite. Unfortunately, one could go ahead and ask to find solutions of similar puzzles where the shape of the hole in the center is any of the remaining 23 pillows. No.