Dick Powell sings "You Must Have Been a Beautiful Baby" and later dresses in blackface and does a Jolson impersonation.
Penny Singleton (from the Blondie movies) as Maggie's personal maid
Melville Cooper (Sheriff of Nottingham from The Adventures of Robin Hood) as Mr. Richards' valet
Tom Fadden* (the toll keeper from It's a Wonderful Life) as a gas station attendant
Looks like I was wrong about this one. It was Irving Bacon.
*IMDb doesn't even list this one!
Hold That Kiss with Maureen O'Sullivan, Dennis O'Keefe, and Mickey Rooney. Junie Evans (Maureen O'Sullivan) and Tommy Bradford (Dennis O'Keefe) are both working class stiffs who cater to society clientele. Junie works for a couturier and Tommy is a salesman for a travel agency. They meet at a society wedding ball where Junie is helping the bride with her honeymoon ensemble and Tommy is hand delivering cruise tickets. They both pretend to be invited guests and each believes the other, aided by the fact that the absent-minded host also mistakes them for guests. Tommy has memorized countless tour descriptions ("When night falls and the shadows start to lengthen in glamorous old Singapore, one is taken back to the romantic days of Marco Polo...") and can pass himself off as a world traveller even though he has never been "east of the Statue of Liberty". Each is impressed that the other is more down to earth than the society snobs he (or she) usually encounters. They start to date, taking pains not to reveal their real social standing. In searching for a date venue that is inexpensive but still "society", Tommy decides to take Junie to the dog show, where they somehow end up as owners of the Saint Bernard that won Best in Show. Eventually the charade falls apart, and after initial anger at the mutual deception and some payback pranks, the couple plus dog ride off together. Mickey Rooney plays Junie's clarinet playing younger brother, Chick.
Someone in the movie once mentions the "Four Hundred" and once Tommy introduces himself as "T. Van Rensselaer Bradford". This is a reference to elite New York society in the late 1800's. It was said that there were only 400 people who mattered in New York society, and the Van Rensselaer family was part of this. A good article about "The Four Hundred" can be found here: http://edwardianpromenade.com/?p=839. Also in the movie they twice use the phrase "soup and fish" to refer to formal wear (once saying "a tux is no good, it's gotta be soup and fish!")