At one point Moe says "I haven't been this confident since i saw La La Land win Best Picture and turned off my TV" a reference to a gaffe at the Academy Awards where La La Land was thought to have won the Oscar for Best Picture only to find out that Moonlight was the actual winner.
Bart's chalkboard gag: "We do not live in our own pee"
The title of the episode seems to be an allusion to a film entitled "Singin' in the Rain (1952)". It is most famous for a dance number that goes by that same name.
The servile gentleman who has the compunction to say "yes" in every statement, seems to be loosely based on a character portrayed by Frank Nelson on The Jack Benny Program (1950).
WILHELM SCREAM: Fund Bunch bowling team member screams after Lisa insults him.