I tried a lot of different things over the years, from everything in a hamper to precut sizes/shapes. In the last decade I've decided to live by the motto that I can cut down, but I can't cut up and I don't like messing with lots of little bits, so the smallest I keep is 6.5" x something larger than a square. I am, however, a scrap quilter. A wide variety of fabric is my paintbox and I can justify quite a lot! Still you have to be able to find what you want/need.
When my yardage is getting narrower, it gets cut down to a 6.5" strip and those go into my small pieces box. Fat quarters and up go into yardage which is sorted by color or theme. I used to have other sizes but have shipped them all out. I met a very lovely crumb quilter on these boards who makes donation quilts for a variety of causes and I keep a flat rate box at my cutting area and put in things larger than an inch square. Periodically I ship the box to her and she gets to play quilt archaeologist digging through the strata of my scraps! I have a box ready to go but it got stuck at home here with me while we were down to one car -- but now there is more and I want to open it up and stuff more in...
Likewise, I have someone from here who gets my selvedges, sometimes I have some lovely lengths from my pieced backs.
I feel a lot better paying for postage than I do dealing with raggedy bits or throwing them away -- there are many solutions to the problem. Some people just make what i call planned quilts and so don't really have leftovers to worry about, what they have they stick in a ziploc bag and offer to quilt guilds or otherwise donate.