Dewey Cox's audition, at which he sings Dean Martin's hit "That's Amore," is an in-joke reference to Elvis Presley. Elvis showed up at his audition for Sun Records wanting to sing like Dean Martin, but Sun owner Sam Phillips refused to record him until he and three members of the Sun house band started jamming on a blues song, Arthur Crudup's "That's All Right, Mama," which became Elvis' first record.
When we see the Top Selling singles chart showing Deweys fictional hit song "Walk Hard" at #4 the 3 songs ahead of it are #1 You, You, You by The Ames Brothers, #2 No Other Love by Perry Como and #3 P.S. I Love You by The Hilltoppers - all of which were actual hits.
When L'Chaim visits Dewey in prison he begins talking in Yiddish asking Dewey to speak the "Mamehlushen" (literally meaning "mother tongue") explaining that the guards won't understand what they're saying and suggests Dewey go to rehab. Dewey responds in German.