The unusual little car Connie's friends drive is a 1936 Fiat 500A Topolino. 'Topolino' means 'little mouse' in Italian, and the car was very rare in the USA.
Benjamin Dingle and Joe Carter read the Sunday funnies about Dick Tracy and the Leopard Lady. She is not an actual Dick Tracy villain but was made up for the movie.
At the hotel restaurant/lounge, the band playing in the background is, as would not be unexpected, all-female. Also, some of the couples dancing are girl/girl, as there were not enough men to go around; "eight gals for every fella" as Ben Dingle said. So of course, when Connie Milligan is called away to the phone, Joe is surrounded by exactly eight young women.
Joel McCrea originally didn't think he was right for the part of Joe and thought Cary Grant would have been better suited. Coincidentally, Grant would appear in the remake, Walk Don't Run (1966), albeit in the Charles Coburn role.