"Puttin' on the Ritz" was written by Irving Berlin in 1927 and introduced by Harry Richman in a 1930 film of the same name. That was only three years after the "The Jazz Singer," so "Puttin' on the Ritz" was one of the first songs to become famous from a movie. It has been more closely associated with Fred Astaire, who also recorded it in 1930, and sang and danced to it in the 1946 film "Blue Skies." And of course, Gene Wilder and many others — including Judy Garland (see update at the bottom of this post).
Clark Gable's one and only dance scene:
Surprisingly, I couldn't find a complete original clip from "Blue Skies" (there is another at the bottom of this post).
Gatsby is a pretty obvious name for a group that plays "Puttin' on the Ritz"
Here is a better "Blue Skies" clip, though on a split screen:
Audio only (h/t Cory Franklin)