I rip tee-shirts into a long continuous strip, ( it’s not hard) and make a kinda yarn from them for my weaving.
Underwear and socks are harder. They usually end up in Mr. Wrekkers dog kennel.
Once in awhile I see a pair of his skivvies getting tugged around the pen. Gives me infinite pleasure.
That’s what I did! 
Although, I don’t know if it was something they actually needed/wanted. I belong to the local Italian-American Club. We had a guest speaker from the animal shelter. We were asked to bring old towels and blankets. I had some of each but also old bed pillows that I stuffed in the bags too.
My husband uses his old underwear and socks as rags in the garage. Then they get tossed into the garbage.
Old t-shirts are the best! The stuff sold as “cheesecloth” these days is worse than useless. I use a big square of old t-shirt (clean, of course) for any cheesecloth need.
Old socks are great dusters. Use a rubber band to attach to a broom handle, and you can go on a search-and-destroy mission after dust bunnies! The Daughter keeps socks for wrapping cold packs. A sock works better than a towel for that purpose.
Old underwear is disgusting. But they can be cut apart, the bits of cloth are useful for many little tasks. The elastic can be used to tie things together, because who can remember where that damn ball of string is?
Gardening to help the grocery bill? All scraps, especially the big, reinforced seams, can be used to tie up the plants. One nice thing about old undies: they are SOFT.
Bed pillows, when I can finally pry open Mr VOW’s clutching fingers, go straight into the garbage can. I consider them to be actual toxic waste, and one of these days I expect to be hit with a hazmat fee.
Our use once, throw it away mentality has made us all forget about rags. You need to spend some time with a pair of scissors and your worn underwear, but in the end (pun absolutely intended), you’ll have a nice pile of rags that will most likely do a much better job than any modern stuff.
~VOW
Sew them into stuffed animal shapes, then donate them as toys.
“Look Honey, it’s your new teddy bear Mr. Brownspot!”
The animal shelter can use old clothing as cleaning rags, and if you know anyone who makes stuffed animals or toys, stuffing them with old clothing makes them machine-washable and dryable.
Hear hear. Yes, you typically throw rags away after a single use (although my mother, peace be upon her, would actually do occasional washer-loads of dirty rags and reuse them), but that’s better than throwing them away before using them as rags.
I do just directly trash crotch gussets and elastic from old underwear after separating them from the more reusable parts of the item, though.
I love hunting rabbits! I’ll be using this to get the space between the washer and the wall.