I have envelopes within folders. Bits of paper that are pretty small go into big envelopes, then the envelopes get labeled something like "Christmas Card Bits" then I put the envelopes into a expanding file folders also called an accordion file folder. Also in the folder are bigger pieces of paper arranged by type, color or theme. I do the same thing for my stickers and other flat scrap-booking type embellishments. Paper that comes in a stack bound together (like big scrap-booking stacks) I keep together and right next to the file folders on the shelf.
I was thinking about how I store paper, and how my file folders are so big its really hard to close them, there must be a fun way to display them while being super organized. So I went searching on Pinterest, and I found this link
And I love it! I don't think I have the self space for it just now, but when I set up my new craft room after my big move in December, I will totally be incorporating this system for organizing paper. I bet you have the space for it though lol.