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: 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: In the abandoned railway station scene, when the heads on stakes appear, you can see in the first shot, Winston is wearing his hardhat after that there is a split-second shot and it disappears. In the very next shot it reappears.
Visible crew/equipment: When the chairs are flying through the air in the courtroom, if you look closely, you can see the string pulling one of them.