Question about shutter speed on dslr?

I'm curious about exposure/shutter-speed. I've been told that, for example, if you're shooting a person who's standing still with a focal Length around 50mm, that you should 'at least' have the exposure at 1/60th of a second. However, what if you you bump it up to-- lets say-- 1/2000 sec? Would I lose any picture quality? And if so, what kinds of distortions would appear? Or rather, what would make it less appealing than the recommended shutter speed?