My wife has a pair of MTB shorts--I think they area by Trek--that are like what everyone else is describing. Pretty good padding with a mesh type lining. The mesh lining has grippy stuff at the bottom to keep the legs from creeping up. They have floppy nylon type shorts over that. No one knows you are wearing padded, grippy mesh shorts under them.
Another vote for the lycra shorts though. I am learning to care less and less about what other people think of my looks. It's all about me!