The scene where Harry is left behind covered in soot while everyone goes to the dance is a nod to Cinderella.
The song "Sing! Sing! Sing!" used at the dance contest was also used in the iconic dance contest sequence in "The Golden Girls" episode "One For The Money" (Season 3, Episode 2).
Dick and Mary's dance scene at the end of the episode is a nod to the movie "Funny Face" (1957). The song being played is "He Loves and She Loves" by George Gershwin.
Episode title: This episode title is a play on "Shall We Dance?," the title of several songs and movies.