I'll sew with anything 1.5" or larger.
I rarely actively save 1" squares unless I have a specific idea in mind. Currently I'm using a bunch of bitty scraps up in my 365 Block challenge.
Recently I've started using small scraps of fabric that I really love to make cards. I keep a stack of card blanks (available at craft stores) right near my cutting table, and when I have a pretty little pile of coordinating scrappy bits, I stick them down with a glue stick to make a quick abstract card. If I'm in the mood, I sometimes do a bit of machine stitching to further secure and embellish the bits. Makes a great card to accompany a gift quilt, or just general use too.