An offset log cabin block uses pieces as small as 1" x 1.5". I have also made regular LC blocks using 1" strips so the smallest piece in those is 1" square. I may even try my hand at making a mini or two. Did you see this post recently?
http://www.quiltingboard.com/picture...y-t275987.html Now that tiny little pineapple block used some little bits I bet.
Tiny applique pieces can be less than 1" square. I am a saver. If I can get 1" x 1.5" out of it, I save it. AND more importantly, I try to use it. I organize my small scraps in different ways. I have bags of 1" strips sorted by value light, medium and dark. I have snippets of odd size scraps in ziploc bags by color. I have ziploc bags with completed HST in sizes of 1.5", 1.75" and 2". I have lots of 1.5" strips, again sorted by value for scrappy log cabins or to sash a block.
It all boils down to your quilting style and what you like to do. I don't mind working with the little bits so I stash them away. But if you are opposed to working with little pieces then you should pass them on to someone who will work with them.