I am having a similar problem, but not with a spinning beachball - I have a cursor I can move around, but nothing I click on works, I can't switch between programs, and Force Quit combination on the keyboard doesn't work either.
I have to restart the machine. This has happened 4 times since I installed Mountain Lion a week ago. It did not happen before.
When it has happened, I have had Safari open and been doing stuff in Safari. If video is playing, I can still hear the video and watch it - just not click on anything. I don't know if that is related or just a coincidence.
My harddrive is brand new as it was recently replaced by the Apple Store, after a HD failure.
Anyone else with this since the Mountain Lion update?