If it's big enough to be on a core it should be on a rack or in a box. We keep oddball colors in the box with a swatch stuck to the outside. That way they are easy to sort through when looking for a specific color. It also protects the rolls and you can stack them under a table or on a shelf out of the way.
There is a balance between the amount and size of scraps you decide to keep. If it takes you 5 minutes to find a $3 piece of scrap, your tripping over dollars to pick up a dime unless your not busy. Take some time and figure out what that 12" x 6" of vinyl is really worth?
In the past we bought sonotubes to sort by colors and when we got slow we would sort through and throw out the smaller pieces. We didn't have a tube for each color, just reds, yellows, browns, blues, metallics... That made it easier to find. We also used the tubes to put finished work ready for installation or pickup in as well. Gets everything off the counters and out of the way so production doesn't have to keep moving stuff around to make space to work.
All the scraps that we threw away would go into a special trash can that school art teachers would come pickup from time to time. Line it with a trash bag. When they show up pull it out and hand it off, or take it to your kids school when it's full.