"Singin' in the Rain": The Hollywood Revue of 1929.
"All I Do Is Dream of You": Sadie McKee.
"Should I?": Lord Byron of Broadway.
"Beautiful Girl": Going Hollywood.
"Would You": San Francisco.
Doing It for the Art: Gene Kelly was sick and feverish the day they shot "Singin' in the Rain" but he powered through and sang and danced at full capacity in a noxious mix of pouring water and milk (added to make the water pop better on film).
Executive Meddling: The studio felt there had to be a musical number in which the titular song was sung. Hence, the Singin' in the Rain sequence with Gene Kelly literally singing in the rain.
For Debbie Reynolds, who was 19 when this movie was made. Invoked by the studio, who felt she was ready to be pushed to the next level of stardom. As Singin lore reveals, it was a bumpy ride for her.
Despite the fact that Reynolds and Gene Kelly's characters (eventually) became sweethearts on screen, in reality the two actors did not get along very well and frequently bumped heads. In fact, Kelly, in his role as director, once mocked Debbie Reynolds' dancing to the point where, after shooting finished, Fred Astaire (who was visiting the set that day) found her huddled under a piano, sobbing. He helped her get her dancing closer to Kelly's draconian standards.
An early draft of the script featured the title song as a number for all three of the leads, not just Don.
An unfilmed alternate ending reveals that Don and Kathy eventually married, and continued acting in movies together. It also reveals that Lina married Cosmo, and landed movie roles that wouldn't require her to deliver a lot of spoken dialogue (such as an unintelligible Jungle Princess).