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On what Stanley Donen and Gene Kelly's Singin' in the Rain (1952) achieves that makes it the consensus greatest Hollywood musical.
Singin' in the Rain was made at MGM's Arthur Freed unit, which was the most sophisticated production apparatus for musical cinema that has ever existed. Freed produced films by Vincente Minnelli and George Cukor in addition to those he made with Donen and Kelly. The unit's films are the apex of the genre.
What makes Singin' in the Rain specifically great, rather than just excellent, is its subject: the transition from silent cinema to sound, which the film treats with a farcical honesty that MGM's films about itself were usually unwilling to provide. The film acknowledges that sound destroyed some careers and exposed the inadequacy of others, and it does this while also being a celebration of the medium.
The 'Broadway Melody' sequence — which interrupts the film with an extended fantasy number that has nothing to do with the narrative — is a formal statement about what musicals can do: they can stop telling the story and exist as pure spectacle, and this can be more honest about what musicals are for than the story they interrupt.
The title number is used as a metaphor for joy, but it's worth noting that Kelly chose to perform it in the rain after the rest of the cast had gone home. The isolation of the number — a man alone on a deserted street — is part of what makes it feel true.