(I know you can get nicer-looking ones but just wanted an image to show what I was talking about) -- that would probably hold the bulkier puzzles & you could always do a traditional puzzle rack for the others that would fit.
We just stack ours, flat ones on the bottom, then the raised ones w/o knobs, then the big-knob ones (or play boards with lots of little knobs), then the small-knob ones on top. It's not a pretty solution, but other than tossing them all in a drawer or basket, I don't know what else to do. (And a drawer would probably be messier than putting the pieces together and stacking neatly...)
I have two sturdy shallow baskets that our puzzles go into. I keep only two or three puzzles out on the playroom shelves at a time and the rest are stored neatly with pieces in them in the baskets in the closet in the playroom. I change the puzzles out about once per week.
We put them in the large ziplock bags. (If you didn't want plastic you could easily sew rectangular bags and do velcro or a zipper at the end.) They can then be stacked, put in baskets, on shelves, etc. Our library puts in a hole with grommet and then hangs them from hooks.