The musical cartoons in the so-called "Golden Era" of Hollywood animation were often "one-offs" rather than ones featuring popular characters and are unfortunately rarely seen nowadays. There was a Woody Woodpecker using Figaro's famous aria from "The Barber of Seville".

Warner Bros. owned the rights to lots of popular songs, and they used (or "recycled" them) in these musical one-offs. For example, there's a short that includes snippets from several songs from the film "42nd Street", which would seem to have been from right around the same time.