When I think mods, I typically also think PC gaming. But people do mod games on consoles too -- and it looks like modders are starting to sink their teeth into Destiny.
Right now, it looks like modders haven't figured out how to do anything particularly game-breaking yet. The main mod I've seen floating around is one that lets modders shoot indefinitely, without ever having to reload or get more ammo. It's an infinite ammo/no reload glitch, as you can see above in the video by 0neThatsExiled.
Other people seem to be messing around with the mod too:
Hopefully things don't get too out of hand with Destiny mods, given that it's a multiplayer game. We'll keep you updated with anything we learn.