I save everything larger than a 1" square. I have always done this, even before I took up quilting. I guess I knew where my true destiny was. :D I have a 3-tier system of organization: scraps, which is anything larger than an index card; strips, anything longer than it is wide and at least 1" wide; and crumbs, which is everything else.
I also save most batting scraps larger than an index card.