1" strips are the smallest I save. I have made a lot of scrappy quilts. I have containers to store them in.

O.K. Now you are all going to get a good laugh. I actually ran out of fabric on one of my first quilts so I sat and sewed up some of my scraps. They are trimmed to 2 1/2" squares and most of them have 8 pieces in them and one or two have 10 pieces. (All green). I don't actually know what I will do with them - it will have to be a wall hanging as I don't think they would take much washing. I do like small and am playing with a few ideas for after I have completed several quilts. One wall hanging I saw pictures of (it might even have been on this site) was of fall trees and the foliage was like fabric confetti in graduating golden hues. So I am not sure where my minimum is yet either but I have saved anything over 3/4" by 3/4". Give yourself some time before you give them away.

Could you explain how you make fabric? I'm curious.

2" is the smallest I will save. I sometimes regret it, because I have tiny pieces waiting for a rainy day that hasn't happened, so I might bring my saving size criteria up the 2.5" or 3".

The size scraps you save should be based on the kind of quilts you make. If you like applique really small can come in handy ( I never know when I’m going to need an eye, or a berry) if you like string quilts long, skinny strips are saveable. If you ever make kennel quilts ( or pillows, padded items those slivers, tiny pieces can be used as stuffing. )
anything You think you will never use you can pass on. I never send my scraps to a landfill, there is always someone willing to pay postage for what I have no use for.

I do have a modest size bin labeled crumbs and I only throw colorful pieces in that I might use later on.

You can make crumbs........just a bunch if scraps sewn together haphazardly, then cut into whatever size square you decide on - like "making fabric". A good no-brainier if you just wants to sew with nothing really in mind, and soon you will have a good stack of blocks to play with, and a few less scraps!

I have a 1 small bin that has pieces about 1"- 2" square, I love to do applique on my embroidery machine and I am always using very tiny pieces for that. I keep pieces from 5" square to fat quarters together in another bin and everything bigger is yardage.
Smaller than an inch goes in the garbage and I keep my tiny scraps limited to that one bin.

Now, themadpatter, this only applies to me. I don't save anything smaller than a 2.5 X 2.5 Over time I have learned that is the smallest I ever use. So, what keep the little stuff that will not be put to good use. I do have a quilting guild friend who makes 1" postage stamp quilts so I donate scraps to her but many I just toss the little ones out. I won't use them and it is the special few who use smaller scraps. Three cheers to those who use the itty-bitty scraps. Love my 2.5 X 2.5 and use LOTS of them! WE all have our passions, right?!

Always an interesting question. When I sew I let my scraps fall into a basket beside my cutting table. When it gets full to overflowing (like it is today) I sort it out into piles by color. I trim the pieces of odd slivers that came about as I cut elements of the previous projects. Then I put them in plastic bags by color and store them in drawers. This gives me maximum opportunity to choose what I need. For a long time I cut them into strips and squares-only to find that I needed something 'bigger' and had to cut into my regular stash storage. I encourage my quilting friends to come and see what they can find and use. I might add that when it comes to batting I keep each type and brand and type separate. I cut all scraps into the largest square or rectangular shape and sew them together with zig-zag stitches and make large pieces which I use in other projects. I do not mix them at random because each type behaves differently after being washed and I don't want that sort of look.