I make a lot of quilts for homeless families (425 delivered since 1996 and counting). At least half and maybe two-thirds of the quilts I make are string quilts. I therefore cut my scraps into strings which could be as narrow as 1.5 inches or as wide as 3.5 and maybe 4.0. Lengths will vary. I do l like the look of the string quilt when some of the strings are pieced from crumbs, so I keep crumbs. I keep strips from one fabric together to scatter in the same quilt. I store these in a very large laundry basket under my machine's cabinet where I can easily reach them. The dark strings are on the left, the mediums in the middle, and the lights on the right. I do not sort by color as I've found that most everything goes together so long as I give the eye places to rest. The strings are all trimmed and pressed when they go into the basket, so if I am careful not to wrinkle them, I do not have to press each seam as I go except with a wooden presser. This saves a lot of time.
If I have scraps left and there is coordinating yardage, the scraps - and any fat quarters - are stored with the yardage. More than once, I needed just a bit more, and the scraps have saved the day.