Chucks for squats, chucks or bare feet for deads, OL lifting shoes for ATF squats (if you have trouble keeping balance, I don't), and Nike Triaxx or Excelerator Runnign shoes for everything else.
i just got some new balance 805's which seem to be way stable in the bottom, they are designed to be firm and prevent pronation. i blow out nikes like it is cool. usually after a month of leg work and they feel spongy at the bottom. no support in the foot anymore. these new balances feel pretty firm, hopefully they will stay that way. a friend of mine got some addidas OL shoes,at tom platz's recommendation, and she seems to like them. the heel might be nice if going ATF as spatts said.
I may be in need of some new shoes for running, tire flip, sled dragging, etc...something that is comfortable, will hold a HEAVY Super Yoke, comes in 16's, and has good traction!!!
I wear Lugz boots when doing seated calves. I find the support a little better than running shoes not to mention the soles on the running shoes start to wreck from the foot rests on the calf machine.
Basketball, turf (football etc), and indoor soccer shoes are also good in that they are very low, flat, stable, and supportive.
I think just about anything in any of those genres or wrestling will work; check out eastbay and look for the cheapest shoes in any of those categories that they have in your size.