Well, the idea would be to push off of the most stable platform possible. That way there is no weird roll, or compression when you push. Converse Allstars are like the old standby for lifting shoes. The only downfall is that there is no heel in them. That may or may not be an issue, depending on your build. I get around in mine ok. The rubber is reasonably hard and doesn't compress much (can't imagine playing sports in them, but...)

For once, cheap running shoes probably are the better choice. I'm sure that they didn't spend much on various density foams to adjust the foot roll or heel strike.

At the end of the day, if you can afford Olympic weight lifting shoes, those are the best for squatting. You will never feel so solid as you will with those on, but they wouldn't be good for deadlifting as you would want to be more flat footed.

It is all splitting hairs, I'm sure. The average trainee can lift in any old athletic shoe and be fine.