It's a bit complicated, actually. The song "Rudolph the Red-Nosed Reindeer" was written by Johnny Marks in 1949, based on a poem written by Robert L. May in 1939. So while the song itself is not copyright-protected, the characters and story it tells are protected by copyright. The copyright for the song expired in 2003, but the copyright for the story and characters is still in effect until 2078. So while you can sing the song without permission, you can't use the characters or story in any other way without