Continuity mistake: Near the end, the Ghostbusters make the Statue of Liberty smash the window on the roof of the art museum before abseiling in. The window gets completely destroyed, but just after Vigo says 'Now we become one' we see the window again and it appears less damaged than before.
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.
Continuity mistake: When Dana and Oscar are at Venkman's place right after the scene where the bathtub attacks them, there is a slight mistake. Dana is shown from her left side, and she is holding Oscar. Oscar turns his head towards the camera, and almost immediately the view changes, and Dana is shown from her right side. Only thing wrong is that Oscar is still looking at the camera without ever turning his head.
Continuity mistake: In the museum, when Dana is trapped and the Ghostbusters lay on the floor, right before Vigo grabs the baby, Vigo's hair is fluffy and messed up. A shot later it's straight and neatly brushed.
Continuity mistake: When the slime is covering the museum an aerial view shows its surroundings empty. At the end of the movie, thousands of people gather around the place, yet the aerial view when the (superimposed) slime explodes is the earlier one, making it look as if the crowd had vanished, only to reappear a second later in close-ups.