(at around 1h 25 mins) The Statue of Liberty is shown at one point wading through water up to her chin; in fact, the waters of New York's Upper Bay are a maximum of 45 feet deep. Given that the Statue is 111 feet tall from top to toe, the water should have been no deeper than her waist, but the line said in the statue's head gives how this can be the case when we hear Venkman say "Keep kicking Libby," indicating that she was swimming, not walking, although the sound of the footsteps is still obviously heard.
(at around 58 mins) The "City of Albany" ghost train, which is said to have derailed and wrecked in 1920, consists of a late nineteenth-century locomotive and cars. Such equipment was still used by railroads in 1920, but a "name train" would have used modern, up-to-date equipment.
(at around 55 mins) Where upon arriving at Peter's rendezvous with Dana, Raymond and Egon proceed to inform Peter of their intentions of going underground. During this dialogue and an over the shoulder shot of Egon, we can clearly hear Ray speak to Peter while the only mouth that is moving in the shot is Egon's.
(at around 1h) When Ray, Egon, and Winston (Ernie Hudson) go down into the sewer to examine the slime, and Winston is using a measuring device to see how deep it is, when he is being pulled towards the slime and Egon and Ray are trying to hold him back, you can hear one of them call Winston "Ernie" by mistake.
(at around 1h 17 mins) When Ben Stein informs Hardemeyer that there is a covering on the main Museum, and that they "can't make a dent," the picture is briefly turned towards the camera, and is obviously a picture of the base of the Statue of Liberty.
(at around 36 mins) When the Ghostbusters leave the firehouse on the way to respond to the haunting of the crystal store, Egon and Winston are wearing tan uniforms while Ray and Peter are wearing dark-gray uniforms. The different-colored uniforms are also shown in Ecto-1 while they're on the way to the store. At the store, however, all four Ghosbusters are wearing the dark-gray uniforms.
(at around 55 mins) Outside the restaurant as Peter and Dana are calling for a taxi, the ectomobile pulls up. The numberplate on the front clearly says ecto-2 this is because the revamped ecto was intended to be called ecto-2 but was later changed to 1a, this scene must have been shot before the change was made.
At the beginning of the film, right after the Ghostbusters II logo, we see several shots of Ecto-1 racing through NYC. In all shots, Ecto-1 is smoking heavily, except for the second shot where the car doesn't seem to smoke at all.
At the end of the film when the Ghostbusters get into the museum the statue of Liberty is standing. Once inside the statue is laid down next the museum but nothing was seen or heard. Furthermore, based on her raised arm the statue alternates between laying facing up and face down between shots which conclude the film.
(at around 2 mins) The carriage with Oscar stops on First Avenue and there are shops in the background. (at around 20 mins) When the Ghostbusters are digging in the street there is now a parking lot in the background.
(at around 1h 30 mins) Near the end of the movie, when Vigo moves the boxes and other items to reveal where Oscar is hiding, if you look at the left side of the screen, you can see part of someone's arm popping out from behind the pillar.
(at around 1h 16 mins) During the "Flip City" song montage, when the large monster ghost is coming through the arch (at the time done with stop-motion animation), if you look to the right of the screen at the top of the arch (in the sky next to it) you can see where the green screen the puppet was shot against wasn't completely edited out.
(at around 1h 19 mins) When the mayor is looking out the window from his office, the eclipse appears to be over Central Park as viewed from Central Park West. The Mayor's office (City Hall) is downtown, so a view of Central Park would be impossible from there.
(at around 23 mins) When the Con Edison Supervisor and the police man arrive where Egon, Peter, and Ray dug the hole on First Avenue, there is a subway station exit with red lights atop it. There is only one subway station on First avenue, which is the First avenue station on the L train, and all of the exits to that station have green lights atop it.
(at around 1h 17 mins) The Titanic has a huge, almost circular hole on the hull. This is incorrect as the ship received a 300 feet long gash below the waterline as a result of scraping along the iceberg. (The ship was also broken in half, a fact rarely acknowledged by movie depictions before the mid-1990s, but it can be assumed that ghost-magic takes care of this problem.)
(at around 15 mins) During Oscar's examination, Egon suggests they perform an "Apgar score" test on baby Oscar. However, Apgar score is a test made on newborns only (usually 1 and 5 minutes after birth), so such a test would be entirely irrelevant in Oscar's case. This is a gag to show that the guys don't know what they're talking about.
(at around 39 mins) When the toaster is dancing, Ray puts the slime in the toaster (on top of anything that would be inside). When the toast pops out while the toaster is dancing, the toast looks like toast, without any sign of slime on them, however, we do know that he moves the toaster to another location, so there's no reason he could not have put bread in and pushed it down before playing the song.
(at around 1 min) Michael P. Moran is credited as playing "Frank the Doorman" even though the character specifically tells Dana, "I'm not the doorman, Miss Barrett. I'm the building superintendent." But this is simply a joke put in to acknowledge what the character of Frank had said at the beginning.
(at around 1h 26 mins) When asked if she (Lady Liberty) can go faster, Ray denies it because the vibrations could damage her. Presumably, she got off the pedestal on Liberty Island very slowly and carefully, using all four limbs.
(at around 1h 26 mins) When Louis runs to the bus stop, you can see the bus in the background waiting for the cue, but, given the way buses run and the fact that his run to the bus stop didn't take that long, it could rightfully be assumed that the bus was pulling up to the stop at the same time, needing to slow down.
(at around 1h 28 mins) After the Ghostbusters smash the glass and metal dome on the roof of the museum, the entire dome wasn't actually totally destroyed. When the Ghostbusters rappel down, the dome is still halfway intact, and later on, when Vigo looks up at the ceiling, half of the dome is intact.
(at around 1h 28 mins) When the Ghostbusters drop in to the museum, Ray and Winston both have lights on their "slime throwers". These lights switch from on and off between many shots until the end. However, slime was not a constantly thrown thing, we know the proton packs have activator switches on the wands, so why not the same with the throwers. In other words, on when they're being used, but switched off to conserve power.
(at around 16 mins) When Ray and Egon are "examining" Oscar, the tape recorder Ray is holding jumps between his left and right hands between shots. However, this could easily be the result of him resting his elbows on the table and putting the recorder down and picking it back up several times.
Winston and Peter switch places twice while in the statue. The original order was Peter, Winston, Ray, and Egon. It then changes to Winston, Peter, Ray, and Egon. It then goes back. However, given that there is enough room up top for them to move around and look at things, and given the apparent time difference, it's entirely probable that they could switch places.
During the "We're Back" montage, as Ecto-1 pulls out of the firehouse with Louis Tully running behind, there are clearly piles of wood and metal leaning against the back wall. During the rest of the movie and in the first Ghostbusters movie when Ecto-1 is parked in the main level of the firehouse, Janine's desk and Peter Venkman's office are at the back of the building. But, given that we know there was probably a space of time between the initial movie, where there was a huge hole punched in the floor, and this one, it's likely they had those materials to rebuild the firehouse and expand it.
People still don't believe the Ghostbusters that they found slime in the underground at the trial many days later. But in order for the real electricians to get the city lights working again, they would have to go underground to do all the repairs in the exact location where Ray caused the damage, and they would have discovered the river of slime for themselves.
It is established that no one believes in the Ghostbusters or the paranormal. However, in the first movie it clearly showed the citizens of New York being terrorized by ghosts, and witnessing the events of the marsh mellow man, and the supernatural forces in the sky above the skyscraper where Gozer is entering Earth. Reporters were even shown filming these events but just five years later and no one believes in them anymore.
Vigo possesses the Museum curator to capture Oscar for him so he can get out of the painting and into the baby's body. But despite the Ghostbusters retrieving the baby when they arrive at the museum and preventing the possession, Vigo is able to exit the painting and retrieve him himself anyway. Since Vigo is gaining power the closer it gets to Midnight when he can actually transfer his soul into Oscar, it's not far fetched to believe that this combined with his actual near possession of the baby could've given him enough power to briefly escape the painting to defeat the Ghostbusters.
(at around 1h 28 mins) When the Ghostbusters arrive at the museum in The Statue Of Liberty, they descend from the crown on ropes. However, the windows on the crown are not large enough for them to fit through on their own, much less with their equipment packs.
(at around 1h 24 mins) When the Statue of Liberty is making her way through the harbor, the water is incredibly calm. Something that big moving through the water would make a large wake or at least some ripples.