Deliberate mistake: When the kids are about to jump start the trap Phoebe hides behind the school bus door. Half a second later, when the trap opens, she's nowhere to be seen, not even jumping or stepping inside the bus. Then half a second later she has managed to get inside, turn around and protect her head while the bus windows burst. It all happens in real time for their hairs are still fluttering. It sure paces up the scene, but it's really awkward.
Audio problem: During the Walmart scene, it seems they edited in Gary saying "Oh, God." However, it's over no visible mouth or throat movements. (01:15:17)
Continuity mistake: When they are chasing the 'muncher' ghost down the street, the light bar is only seen flashing on the Ecto 1 when seen from inside the restaurant. The rest of the time they are not on. (00:59:00 - 00:59:30)
Continuity mistake: The mini Puft makes a hole in the marshmallow bag between the letters H and M. In further shots the hole is fixed and it's now between the A and the L.
Continuity mistake: While driving around the city, Trevor wonders if the ghost wasn't really a racoon. Podcast has his ecto-googles on his head, but a shot later, when Trevor suddenly stops, he has them on his eyes.
Visible crew/equipment: When Trevor is uncovering the ECTO-1, you can see the camera's shadow on the left of screen and even briefly on the car front.
Continuity mistake: When we first meet Phoebe she is fixing a plug, body tilted to the right, left arm raised. A shot later her body is tilted to the left, left arm in a different position.
Other mistake: When Gary is about to test the trap check the bus door on the right: there's a reflection of what seems to be his shirt, but it's a long sleeved wrinkled shirt, not the short sleeved he's wearing. What's even more bizarre: when Gary moves around, the reflection stays completely still and, drumroll, a shot later, from the very same angle, the reflection disappears. Either it was a crew member hiding, or a CGI-made reflection placed to hide some unwanted crew or equipment reflection.
Continuity mistake: When Gary is testing the trap on the car, he puts his hand very close to it. A shot later it's further away, and he's reaching for the trap again.
Continuity mistake: When Dana asks Venkman if he marked the cards her right hand is away from the electro-shock generator in the front shot, but suddenly on top of it in the angle shot from behind.
Continuity mistake: When Gary and Callie are having lunch, there's a waitress behind him that passes next to him. When she's next to Callie, shot changes, and she's now a meter behind, approaching Callie again.
Continuity mistake: After Gus drops the trap his body is closer or further away from it and his right hand is close or away from the trap. This changes back and forth all the time.
Continuity mistake: When Grooberson drops the trap in front of the kids, the handle points at Podcast. A shot later it's in a different position, parallel to the kid. Also, the cable's position changes several times.
Continuity mistake: When Gary and Phoebe are trying to open the trap the two cables hanging down the fender and bonnet are either close together or apart, depending on the shot.
Continuity mistake: When Gary shows the kids a YouTube video of the Ghostbusters, he points his finger at the screen. From the opposite angle his finger is away. Then back to the wide-angle, and he's retrieving his finger again.
Other mistake: In Walmart, some mini-Stay Puft Marshmallow Men turn on a propane grill and set each other on fire. Gas grills on display never, under any circumstances, have propane tanks installed. This would create an obvious safety issue because there is nothing keeping a customer from accidentally (or deliberately) starting the grill and causing a fire. Additionally, propane tanks are never stored inside for similar safety reasons.
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?).
Suggested correction: There's nothing in the movie to indicate that people in general have "forgotten" or "dismissed" the existence of the Ghostbusters, nor is that the "whole premise" of the movie. The fact the teacher is a fanboy and that the characters literally watch old news-clips and commercials for the Ghostbusters kind of goes against this. People simply just stopped talking about them because they did their jobs too well and went out of business 30 years prior... they were no longer relevant. I mean, if you want a real-world-analogue, just look at 9/11. It was a massive, generation-defining event, and yet outside of brief memorials once a year (which honestly, fewer and fewer people seem to pay attention to every year), people basically don't talk about it at all anymore. The only characters in the movie that don't believe in ghosts/the Ghostbusters at first are the kids. And their mother has been purposely sheltering them because she hates their grandfather-a Ghostbuster. So it makes sense they wouldn't necessarily know about them.
9/11 was a different kind of event; it didn't have 4 easy to remember heroes who already were on magazines covers all over the world and while it certainly dropped off the radar in many ways, some consequences in the long term have been permanent and it is in the history books. Here the world had proof that there are other dimensions, the dead, etc, and years later the Ghostbusters are relegated to a few youtube videos with a few thousand views (that with Peter supposedly teaching advertising and promotion, even). I didn't mention the kids, although the movie itself knows it's absurd that Podcast does not know anything about it and there's a joke about it. I understand if someone makes a point about the movie taking an ample creative license for the sake of not having to deal with 'realistic' implications of its comedic prequels since it wouldn't service the kind of story it wants to tell here, but I am surprised you say that the Ghostbusters here are not forgotten or dismissed. Somehow they are so fringe that not even the conspiracy theory guy knows about them, and the teacher knows because they are a childhood memory.
Like Ray tells a young Jason Reitman in Ghostbusters II, "Well some people have trouble believing in the paranormal." The public would have even less of a reason to believe in or think about the Ghostbusters since there were no Ghost sightings in thirty years. Not to mention the fact that men walked on the moon six times between 1969 and 1972 and astronauts were viewed as heroes, but we haven't visited the moon in fifty years, and astronauts are no longer regarded as heroes.
We keep conducting research in the field sending people in space when and where necessary and people are well aware that astronauts exist, even if they declined in popularity. It's not random obscure knowledge you can get only if you are looking specifically for it on some Youtube channel that a science nut and a conspiration theorist never heard of before. And we are again comparing something that does not have the same impact it would have to learn that dead people still walk (so to speak) the Earth. BTW, I am not sure (but I could be wrong here and please correct me) that the movie says that there have been 'no' ghost sightings at all; Ray said that they received less calls, not enough to pay their bills, not that ghosts disappeared entirely. It's just that in the Ghostbusters universe, people are kinda jaded about everything, which worked when the movies were comedies and you could say it was obvious paradox and satire that they would save the planet and still get sued once they weren't relevant anymore.
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.
Suggested correction: Most likely, Egon's ghost returned the PKE Meter to the house, knowing how important it would be to help fight Zuul.
Continuity mistake: At Walmart, when Gary finds the sentinel his eyes are looking down. There's a close-up of him slowly looking upwards and finally staring at the creature. Shot changes, and he is looking down again and repeating all previous movements.
Visible crew/equipment: When Gary opens the trap next to the 2 school buses, he is wearing a cable attached to an earpiece on the right side of his head. The kind of cable actors wear to receive instructions from the crew. There's no reason plot-wise for him to be wearing it (he isn't receiving any instructions from the kids, who are right next to him).