OK, I just watched the movie to try to answer your question! He doesn't actually sing any songs, but there is a song playing in the "background" as he dances to it on the stairs. That takes place immediately after Billy's father hears and agrees to Billy's plan to take each grade over again for two weeks apiece. The song is by Culture Club, and the title is "I'll Tumble For You," sometimes written as "I'll Tumble 4 Ya". Let me know if that's the one you were referring to....