Basically, the only competition for The Band Wagon (TBW) for the title of MGM's Uncontested Musical Champion is Singin' In The Rain (SITR). SITR is certainly the better-known of the two and features one of film's iconic dance sequences. That said, in an itemized comparison, TBW is the winner and by more than a nose.
Star: Astaire over Kelly. As usual. (Stop that. You know it's true. There's no point in contesting it.)
Co-star: Cyd Charisse slam-dunks the ever-annoying Debbie Reynolds (herself a major overall factor in SITR's loss to TBW), but herself gets trounced by Jean Hagen (best squeaky virago ever). A draw.
Supporting: Jack Buchanan, Oscar Levant, and Nanette Fabray triple-team Donald O'Connor (in his defining and best screen performance) and edge out a victory.
Choreography: Kelly and O'Connor (under Kelly's direction) hoof up a storm, just outdoing Astaire and Charisse. Kelly's title track number over Astaire/Charisse's Dancing In The Dark, although the latter is the best duet Astaire's done since Night and Day with Ginger Rogers, and is itself a textbook to economy, style, and subtlety in choreography.
Music: Dietz and Schwartz over Freed and Brown. Despite Louisiana Hayride being a kitsch feature almost unparalleled in MGM musical history.
Book: Comden and Green's TBW ties Comden and Green's SITR.
Finale: Girl Hunt Ballet over Broadway Rhythm Ballet. By a mile. The former is tongue-in-cheek, visually-engaging despite its smaller scale, and imaginatively-choreographed. The latter a bit obvious and, in typical Kelly style, a little heavy-handed in its inclusion of "serious", "dramatic" elements. Cyd Charisse too, too hot in both, of course.
Overall: SITR lights up in the Hagen-Kelly sparring and O'Connor's zinging bits, as well as in the "early days of talkies" bits and the proto-postmodernism of the Beautiful Girls sequence, but Debbie Reynolds and the love interest numbers cause the film to drag and buckle. TBW, by contrast, hums throughout, going from strength to strength: Buchanan's God of the The-ah-tuh role, Levant as Levant, Astaire's wistful By Myself, Triplets (why no Hoops?), the disaster that is the Damnation Scene, Astaire and Buchanan, signifiers for "class", soft-shoeing on I Guess I'll Have To Change My Plan. An uninterruptedly-enjoyable movie, from start to finish.
So, there you have it. See The Band Wagon. Become a TBW fanatic. It's that simple.