I keep all scraps. Stringy pieces get put out for the birds in case they can use them for their nests. I've never seen a nest with my strings but the strings do disappear.
Sometimes the girl scouts have a paper making project and I let them use the scraps to pick out what they hope to end up with in their paper.
I don't sort by color and it goes into ziplocks ultimately but I have a round oatmeal box that I drop things into as I am cutting and quilting. Light weight and can be squeezed into small spaces easily.
I like RST's idea of only one bin full of scraps before it is time to get rid of them. I'll donate the bigger scraps to Goodwill or Salvation Army if my newly designated bin gets too full....