My guess would be singing is not considered polite, unless the host is encouraging it, or it's somehow part of the program.
My sisters-in-law both love singing, and often used to sing at slightly inappropriate times. The story has it that my grandmother-in-law eventually had to make a strict "no singing at the table" rule for the house, to prevent my sisters-in-law from singing so much.
My family has, for the most part, pretty terrible voices, so none of us sing at the table, or anywhere else!
As far as "other ways of amusing yourself at the table", almost all of the ways I know how to amuse myself at the table are considered rude (setting up a goal post and trying to score with various objects, twirling various objects, playing with condiments), so I don't think I'm qualified to answer that portion of the question.