Revealing mistake: When the window slams on Nordberg's hand it is that the window stops right above his hand. When he pulls his hand out the window doesn't go all the way down and a wooden cube is visible on the edge of the window stopping it from closing. (00:06:50)
Revealing mistake: During the car chase, the doc's (already busted) car collides with a missile transporter. The explosion occurs just before the car impacts with the vehicle, and looks more like the burst of a flame thrower than a real explosion. (00:34:25)
Revealing mistake: When the Queen throws the first ball at the baseball game, if you look closely you can see the string which made the ball do its loops. (01:03:30)
Revealing mistake: Jane stands up and strips supposedly naked in front of Frank on their first date. In the widescreen DVD, it's obvious she has a skin colored top on.
Revealing mistake: Frank's plane is labeled near the front for the fictional Atlantic Overseas Airline. When Frank is speaking to the press, the plane's real TWA logo is still visible on the tail.
Chosen answer: "I'm Into Something Good." It's the second song listed in the music credits, and if you don't read all the credits you miss a good part of the gags in the movie.
"I'm Into Something Good" is played during the scenes where Frank is going out with Jane. The song the person is asking about is this song. Https://www.youtube.com/watch?v=wn-MlISDsNs.