Fair for Its Day: The rather incongruous song and dance sequence in the poor black ghetto near the racetrack. Nowadays we think, "They're treating them like performing pets!" But in 1937, it was actually pretty progressive to employ dozens of black actors in a major song and dance production. They were at risk of having the film refused by movie theaters in the South.
The second half of A Day at the Races has an extended musical interlude which starts with Allan Jones singing "Tomorrow Is Another Day," which is followed by Harpo using his flute to summon a black chorus which sings "Blow That Horn, Gabriel" and "All God's Chillun Got Rhythm." (The chorus has nothing else to do in the movie except reappear to sing the finale.) Many Marx Brothers consider this sequence as objectionable on an Ethnic Scrappy level, but it's not really that bad by itself and the choir are very good, but it just stops the plot dead and its earnestness clashes painfully with the Marxes' usual slapstick and wisecracks.
There's also the water ballet sequence, which even the film historian on the DVD commentary advises you to skip!
Values Dissonance: The whole sequence with the brothers pretending to be black is mighty uncomfortable today.