I save everything, but don't cut it apart any more because I needed a piece of a fabric that I had left from a quilt to replace part of a block (big oops with DGS) and I wound up having to piece some squares together then cut the HST.
I press my scraps and lay them flat in containers by general size. I only trim tails that are narrower than 1/2" (I swore I would never applique and found myself making a Halloween quilt and needed really small pieces for my cats and rats and had to cut up a 3" square when I only needed 1" squares of fabric.