The songs that Val Kilmer sings in the film are actually performed by him and were featured on the film's soundtrack released in 1984 under Kilmer's character's name Nick Rivers.
When Nick is leaving the East German checkpoint by train, a voice can be heard over the PA saying, "Der Zug der jetzt auf Gleis 3 steht... hat uns alle überrascht!", which translates to "The train that is now on platform 3... surprised us all!"
When Nick Rivers is in jail, a head shot of Cher can be seen on the wall of the cell. Val Kilmer was dating Cher at the time.
The waiter who asks if Nick is ready to order, ostensibly in German, is actually using the Yiddish curse "Go take a shit in the ocean."
When Lucy Gutteridge looks down from the balcony onto the street to see hamsters and mice, it actually was a miniature from Superman (1978). David Zucker, Jerry Zucker and Jim Abrahams stated in their DVD commentary that they found it in the old Shepperton studios and thought it would be a great idea to use it in somewhere in the movie.
Jim Abrahams, David Zucker, Jerry Zucker: as the German soldiers who show up at the door of the prop room to arrest Nick.