I cut mine. I cut as many 6.5 inch squares as I can, then 4.5 inch. Then I do strips 6.5 inches long by whatever width I can get between 1-3 inches. If anything is left of any size after that, I do 2.5 inch squares.
Anything left after all of that is still kept...I will randomly sew together little scraps to make little stuffed balls or "mice" for my cats to play with; I just overlap the fabric and zigzag each seam and then more random zigzagging to catch everything, no hidden seams, no pre-defined shape really, and the smallest scraps are used to stuff them. The cats don't care - I store the completed toys in a big jar half-full of catnip and the cats go bonkers for them no matter how ugly they are.
I mostly only toss out very thin strings that have a lot of frayed & stringy fabric hanging off. Or if my scrap bin is gone crazy or if a fabric is just not making me happy I will toss slightly larger pieces, but usually nothing wider than an inch is ever tossed. It just feels wrong to throw any of it away but I can only indulge my inner hoarder so far!