My PSU exploded, but I see no damage, should I plug it back in?

No sane person would actually plug in a psu that had already explored twice. Lighten up.

I don't know what your specs are like but get a good PSU that is at least 80+ Bronze certified and 150+ watts of headroom. I recommend picking up a PSU that uses a single 12V rail but that's just personal preference, but I do highly recommend picking up an EVGA G2 series. Also if you aren't using a surge protector get one, it's worth every penny because it's potentially saving you hundreds if not thousands of dollars. If you have any questions please ask I'm more than happy to help.

Recommended Reads
PSU Ranking- https://community.newegg.com/eggxpert/computer_hardware/f/135081/t/45344.aspx
PSU Reviews- http://www.jonnyguru.com
Surge Protector Article- http://www.cnet.com/news/9-things-you-should-know-about-surge-protectors/