New Balance cross trainers are best. They have rotational compression chambers (excellent for calf raises) and the base of the sole actually flares out for greater stability when doing heavy squats. They are also one of the few trainers to be available in a 4E width (I get my huge wide feet from the Neanderthal side of the family...).
Don't wear running shoes. It doesn't provide a firm stable surface. And all the heavy lifting pre-maturely compresses the insole and all the cushioning features. So if you actually run in those shoes, you're not doing your knees any favors.
Wear a pair of cross trainers or a pair of Converse Chucks like Someguy mentioned above. Cross trainers typically have firmer insoles and better ankle/heel support. Chucks are very basic. The rubber soles are just layers of flat rubber and cardboard. They seem to be really popular with crossfit people and powerlifters.
I try to wear shoes that have a level sole; that is, a sole that has even thickness in the front and back. I do this because it helps keep my weight level, instead of causing me to lean forward like most running shoes. Running shoes are for running.