Don't let the fact that Singin' in the Rain is a bright and shiny musical fool you: It has a pretty sophisticated narrative structure. First of all, we have a lengthy flashback right out of the gate, as Don tells Dora Bailey his life story. The fact that the tale of his sophisticated upbringing is all made-up establishes one of the film's themes immediately: truth vs. perception.
The "Broadway Melody Ballet" sequence toward the end of the film is also an interesting narrative device; it's totally abstract and open to interpretation. The whole thing takes place on a series of stages, so we don't really know where in space—or time—we are. Since it's framed as musical number that Don's pitching to R.F. for The Dancing Cavalier, it can even be viewed as a sort of film-within-a-film.
Speaking of which, Singin' in the Rain already has two other films-within-a-film. Of course we have the production of The Dancing/DuelingCavalier, which is fraught with problems that propel the narrative and help us understand the characters. Then we also have Singin' in the Rain, the movie advertised on the billboard in the very last scene of the movie, which stars Don and Kathy and delivers their happy ending with a big yellow bow.