"The film makes reference to famous police detective Dick Tracy, however the film is set in 1928 and the Dick Tracy comic wasn't released until 1931." The narrator is the one that mentions Dick Tracy. The narrator makes numerous statements indicating that he is recounting events that took place in the past, so the narrator mentioning Dick Tracy is not an anachronism.
Jennifer Grey and Madonna perform a rendition of "I Surrender, Dear", however the film is set in 1928 and the first version of the song wasn't released until 1931.
The first line in the opening song, "Big Bucks," is "The best things in life are free." The audio matches the visual up to "are free", at which point the waiter clearly begins to sing something entirely different.