I cut narrow selvedges (3/4" or so) off and keep those in a jar marked "ribbons" - I use them for all sorts of things around the house - the prettiest ones get set aside to be used when wrapping handmade gifts; I have a bowmaker template thing I can wrap them around, put in a stitch or two to secure and pop it off to make a selvedge bow that I think looks neat. They're also great for tying plants to stakes out in the garden. DH uses them to tie up cords and hoses and things in the garage. I've used them for things like tying around tupperware to make sure the lid won't pop off during travel. All sorts of uses!
I make foundation-pieced crazy quilts, so I will save very small and odd-shaped fabric scraps for that. Anything I deem too small or too frayed goes into a trash can I have lined with a fabric bag and when it gets full I stitch the bag shut and give it to my pets. Batting scraps and thread snips go in there too. I have 3 dogs and 2 cats and they all like sleeping on my scrap-bags. They usually survive about 3 washings and then need to be tossed because they get too lumpy inside; it seems to work out that I have a new one ready to go right about when one is needing to be tossed out.
I think the batting scraps help keep it fluffy - I notice if I have a larger than normal amount of poly batting scraps the bag seems to last longer. I snip up batting scraps - usually my batting scraps are long strips about 4-5" wide, I fold it up and chop it up with my big industrial scissors so it'll mix around better. I don't bother to do anything with the fabric scraps because usually only very small scraps make it into the bin in the first place.
I DON'T put fabric or string out for birds...my largest dog is a lab/husky mix, I just brush her on the back patio and leave the fur there for the birdies. They LOVE it, and there's a ton of it! I don't know how that dog isn't bald, so much fur comes off of her...