I tried keeping scraps cut into "usable" widths but it didn't work well for me. I kept them separated by size in a variety of large zipper bags and small boxes inside a larger box. It was a better solution than my previous one where I used a large wicker laundry hamper for my little bits. That was always a mess of fabric and tangles and I had to press everything before I used it (keep in mind I hate ironing), and I had to dump out the whole thing to find things.
Edit: Smallest I kept was 1.5, went up by inches for 2.5, 3.5, 4.5, and then jumped to 6.5.
No matter what size I wanted, it seemed like I didn't have it. You can pretty much use nothing but 2.5" widths and nothing else for years and years -- some people like that, me not so much. Truth is, even though I have a scrappy style I wanted to do what I wanted to do, and not what my scraps made me do.
Currently fat quarters and larger are kept with the "yardage" in totes that should be sorted by color or collection (batiks, etc.) When I moved I combined several groups into not yet full totes and I need to spend some time reorganizing the totes.
As my pieces get smaller I cut a final 6.5" WoF (or at least bigger than a square) and those go in my one "small scrap" tote. I figure that I can cut down but I can't cut up and it is a good size for me to use and store. They are loosely put in color order but sometimes I'm using all florals or some other theme -- the size is easy to grab a handful and sort through them in batches.
I need to find a new crumb quilter to accept my small bits. I keep a flat rate postal box on my cutting table and everything smaller than 6.5" goes in there. 2" square or smaller gets tossed.