Favorite Shoes for in the gym?

I would suggest getting fitted at a professional running/sports store. Look for individual/small chain stores for the most knowledgeable service..... sounds pricey, but I really haven't seen much of a difference between buying them there and buying them somewhere else.

going the cheaper route - I would suggest chuck taylors and running shoes. More expensive - Go with a pair of Nike Romaleo 2's for squats, chucks for deadlifts, and whatever you like for the rest!

Inov-8's are great for cross training. Stable enough for lifts while still being good for running. If I'm deadlifting heavy singles, I'll wear an old pair of new balance minimus.

For heavy squats, cleans, or snatches I wear lifting shoes. I use Reebok Crossfit lifters because I got a pair for 60$ on sale. If money wasn't an issue, I'd go for Nike Romaleos.

I wear a pair of full leather shoes from an obscure German brand. They're leather, beige, with a crisscross pattern on the front that makes them look like some 80 year old man's shoes. They are flat, with a little bit of a heel lift, and once broken in, they fit my foot like a glove and provide all the support I need to lift in. There is no flex, or spring. They don't match anyone else's neon workout clothes, nor do they match my neon workout clothes. BUT I wouldn't lift in anything else. Well, that's not true. I would lift barefoot also. I tried in regular running shoes once. It was terrible. Don't do it.

I've heard good things about Chucks but you can't rip me out of my Rohde shoes for lifting. Given as they're probably 20 year old vintage shoes that I only acquired because my shoe shop has some REALLY odd stock tucked in the basement, I will be pretty sad when they go.

I don't do 'crossfit shoes'. You need something very different for running. So I have running shoes for if I want to run, and lifting shoes for my lifting. Well, they're not lifting shoes. But they work pretty great for it. I haven't found anything else comparable for the support.

Chucks or wrestling shoes for lifting.

Chucks/Anything with a solid flat sole for Powerlifting

Adidas (Power Perfect 2/adiPower) or Nike (Romaleos 2) or anything with a raised heel for Olympic Lifting

I am currently using Fila Tennis shoes for the gym. The reasons,
the shoes are tough and sturdy,
can manage jerk motions obviously because of the game in reference,
Are flat enough and curved enough at the same time,
Awesome comfort,
good enough ventilation,
great cushiony sole...

But that's my taste

Here is similar to what I bought:
http://www.jabong.com/Fila-Court-Mate-White-Tennis-Shoes-686193.html