"Shame and Scandal" ("Harlem Man") in Val Lewton's film (title escapes me: I Walked With a Zombie?). Sung by a pretty good, very serious singer, not the Duke of Iron...Lord Invader? Or just an actor?
Any number of Gene Autry films of course, with songs mostly in an early-country music vein.

I know more will come to me as soon as I send this off (perversity of memory).