During the play they perform "You're the One That I Want". The song is not actually in the original play Grease. It was written for the film. However, for a small fee, it is perfectly legal to change the music in the play to include songs from the movie.
Kitty (while making a joke about Marley gaining weight) says "Are you playing the lead in Grease, or in Hairspray?", John Travolta played one both the films Grease (1978) and Hairspray (2007) as the leads.
In the play version of Grease, Danny does not sing "Greased Lightnin'." Kenickie sings the lead.
The hand jive scene and the sleepover, complete with Sandra Dee, are spoofs of scenes in the movie Grease (1978).