"Flat Top" is the only actual Dick Tracy villain depicted in the cartoon, but he is never explicitly named or labeled. He is shown launching miniature fighter planes of the top of his head (get it?).
"Was that trip really necessary?" was a public service advertising slogan during the days of gas rationing in WWII. Even seats on passenger trains were better left to soldiers on furlough.
Warren Beatty wanted this animated short to be shown before his movie Dick Tracy. But Disney (who was making and distributing the film) refused.
In the dream sequence, when we see Daffy's silhouette in the door of his office ("Duck Twacy"), the silhouette of Daffy briefly changes twice into Dick Tracy's silhouette.
The line "Love that man!" was a catchphrase on the radio show, "Fibber McGee and Molly".
The taxi that pulls up is labeled "Tom's Taxi", probably referring to Thomas McKimson.
Warner Brothers actually used a picture of Duck Twacy for a pamphlet about non-discrimination against homosexuals. The hat was gray, but so was the hat in the Piggy Bank robbery.