I have my scraps sorted by color and they are folded (somewhat) neatly to minimize pressing. I elected not to try cutting into standard sizes because I don’t normally do scrappy quilts and I thought it might be more useful to have a large chunk of fabric than several small squares. My only exception to that is when I get pieces less than 2.5” wide I will cut those into smaller pieces that are 1.5” or 1.75” wide to use in Pineapple blocks.
I think if you know you will use particular sizes of scraps, you will save yourself time in the long run by cutting them into those sizes while you are already cutting the other pieces.