Other mistake: Pay attention to one of the early innings of the baseball game as one of the Angels players hits a home run. Though probaby intentionally set up this way as to avoid any problems of "logo patents" and such with the real life MLB California Angels team, he is wearing a helmet without the team logo on it. (01:07:35)
Other mistake: While Frank Drebbin is saying the words "dew glistening on the grass" you can clearly see a woman in the audience at the bottom of the screen repeating every word he says. (01:19:55)
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.