Boxes, Bags, Bins

If you are like me, as you've sorted your scraps, you've put them in boxes, bags and bins that you've had on hand. The 18-gallon plastic totes hold a lot!  Large shopping bags do too.  Shoe boxes, computer paper boxes and smaller totes are good depending on the size of the scrap collection.  If you choose clear plastic containers, you can easily see what is inside.  Label other containers with the contents.
My favorite containers for not only scraps, but also projects in progress, are the flat boxes I gather at my local Aldi store (a bulk-type grocery store).  Because of the way I store my scraps, they need to be pressed before I can work with them.  As I press them, I put them into a flat box so they stay nice and flat.  As pieces are cut for a quilt project, they go into their own flat box so they are all together when I am ready to begin putting them together.  I keep a flat box next to my sewing machine.  Pieced items that need to be pressed before the next step can be easily carried to and from the ironing board. And if I get interrupted and the project has to be put away, it goes into its own flat box, ready for me to pick it up and start again. 
(These flat boxes fit nicely inside an 18-gallon tote!  I have a lot of them stored inside MY 18-gallon totes...projects waiting to be completed and already pressed leftovers from previous projects...they will get used eventually!)