Reddit mentions of Origami Insects (Dover Origami Papercraft)

Well, it depends on what kind of origami you want to do.

These days I mostly do modular (fancy word for multi-piece) geometrical origami, and I love models that are simple and robust. Super-thin origami specialty paper works just fine, but I don't really need the level of precision it offers. So I use paper from brightly colored office memo cubes and old page-a-day calendars. Sometimes those pages are a bit off from truly square but they're close enough.

Thin, dedicated origami paper is a necessity if you're going to fold a complicated, single-sheet model like John Montroll's lobster (from this wonderful book) or Robert Lang's ant (from this excruciating gauntlet). Ordinary paper gets too thick and unwieldy with all the layers.

Sometimes the models in books and online galleries appear to be made from paper beyond your generic craft-store sheets, though. Maybe there are high-end origami paper retailers? I can only assume so. Beyond that, I've seen blog posts from people who mention they craft their own paper, which is a dizzying height of showoffery that I shall probably never attain.