I've wanted to get Fallout 3 for a while now, but I heard that it has some problems on the PS3. I heard that the GOTY edition tends to have more issues than just the regular game, is this true? I really only want it for Broken steel, but I have no problem spending an extra $10 on the PSN store to get Broken Steel if it'll prevent some game-breaking glitches.

I can't get it on Xbox since mine died on me not too long ago and I've pretty much given up on it, and Oblivion has been crashing on start-up on my PC lately, so it's likely I'd have the same problem with Fallout

Tl;dr : I want Fallout 3 for the PS3. Do I get the GOTY or the regular edition?

You should be fine if you start a fresh new save file with the GOTY with minor problems. If not then just buy Broken steel.