Plot hole: The whole premise of the movie is that history would write off the existence of the Ghostbusters after the events of the first movie. In that movie there was prolonged large scale destruction in the heart of a city with millions of inhabitants. It's simply impossible that people would forget or dismiss it. And that's if we do not even begin to assume that the second one happened, even if the director said it did; nothing in his movie shows that, and for a good reason (Statue of Liberty, anyone?).
Continuity mistake: Callie finds the 'secret' (even if it's in plain sight, really) basement with Egon's stuff following the instrument she found in Phoebe's room. She has the tool with her, she does not have it with her when she is possessed, but it is back at the house when everyone else is home.
Stupidity: Spoiler: Everything goes well in the end and all the ghosts (and returning demigod Gozer) are sucked into a giant ghost trap. In fact everything goes so well that the trap selectively spares one specific ghost so it can have a lengthy farewell scene, hug and smooch every single character at leisure and leave on its own terms.