I use a file drawer and have letter size hanging file folders. In each one, I have two or three letter-size manilla folders inside; each one labeled with a different color. The colors are in alphabetical order. The only exceptions to both of those "rules" are the Whisper White and Very Vanilla. Both of these colors are in the front and each has its own complete hanging folder.
This makes it a lot easier to find a small piece of cardstock if I need it for a layer or to punch or die-cut something. Here is what the inside of one of the manilla folders look like.
You might notice I do have at least 2 or 3 full pieces of 8.5 x 11 cardstock in each folder in addition to the scraps. I do this because when I go to a crop, I take a tote full of my 12 x 12 pieces of cardstock and then I pull out this drawer and take it with me, too. If I'm making a card, or need to mount a larger photo, I have 8.5 x 11 available and don't need to "waste" a 12 x 12 sheet. Plus it's just great to have all the little scraps at a crop too for punching & die-cutting.
I keep my miscellaneous cardstock - confetti, naturals, glossy white, etc. - in a separate drawer. This drawer is underneath my scrap drawer and I keep the 8.5 x 11 full packs and the scraps of the same color all together. (The scraps are in the manilla folders.)
If you have any other ideas on how to organize paper scraps, please share your comments below!!