We all run random dungeons. Usually, we'll just do our job, tanking, healing or DPSing to get through the dungeon as fast as we can. But every once in a while, we may get the inclination to mess around a bit.
My main character is a disc priest. Since all I really have to do in heroics is shield the tank every once in a while, I'll often run around like a crazy person using Holy Nova to try and beat the DPS. Not only does it make the runs go faster, but it's more fun than just standing around doing nothing. Sure I could use other, more normal DPS abilities, but it's fun to see those golden bubbles everywhere.
If I'm in a group with guildies and they know it's coming, it's sometimes fun to play "how low can you go" with the tank's health. I'd never try it in a PUG random, but with friends who can laugh if it goes wrong, it's fun. One thing I've always wanted to try but never have is healing in shadow spec. While it wouldn't really be difficult, I don't really know how random groups would react to it, having their healer running around in Shadowform all the time.