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.
Revealing mistake: During the courtroom scene, one of the ghosts grabs the prosecutor and she is hanging upside down. Her dress comes down a little, but when she is about to fly through the door you can see part of the dress is pinned to her panty hose to stop the dress from falling off.
Continuity mistake: When Vigo throws the streams from the proton packs back onto the Ghostbusters, there is a wide shot of Egon getting thrown way over to the right. However, when they do a closer shot, he is way over to the left, closer to the other three Ghostbusters.