Elfman was also literally a singing waiter for a time. I can't remember the article, but remember Cassandra Peterson saying that she worked with him at the same restaurant.
Rod McKuen (he with the scratchy voice and so many haunting songs) sang "Jean" in The Prime of Miss Jean Brodie (1969) and also sang "I'll Catch The Sun" in Joanna (1968).
Please don't forget the late Sir Richard Rodney Bennett. He could do it all - beautifully (IMHO.) The anniversary of his passing is coming December 24.