Continuity mistake: In the scene where the Ghostbusters arrive at the museum, and Egon tells Peter what he's found about Vigo, while speaking to each other Peter gets a bag and hangs it around his neck. In the shot where you only see Peter's back you can easily see the strap hanging straight down. But in the shot where Peter's face is visible, he's holding up the strap with his hand quite high. Cutting back to Peter's back the strap is hanging straight down again.
Revealing mistake: When the Statue of Liberty is about to smash the museum's glass dome, the shot outside is really bizarre: The Statue is standing on a park, with a pink cloud covering what is supposed to be the museum. Yet the lower part of the cloud plus the people around it, show that there is no building nor any foundations there: Just a superimposed pink cloud over an empty field.