The "City of Albany" ghost train, which is said to have derailed and wrecked in 1920, consists of a late 19-century locomotive and cars. Railroads still used that equipment in 1920, but a "name train" would have used modern, up-to-date equipment.
Ray, Egon, and Winston go down into the sewer to examine the slime. Winston is using a measuring device to see how deep it is when he is pulled towards the slime. Egon and Ray try to hold him back, and one of them calls Winston "Ernie."
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. It's obviously a picture of the base of the Statue of Liberty.
In the case against the Ghostbusters breaking their restraining order, the court never bothered to have Dana testify. She asked them to help her and Oscar, making her an accessory to them digging the 1st Avenue hole.
Peter and Dana are waiting for a taxi outside Peter's apartment building when the Ectomobile pulls up. The license plate on the front clearly says ECTO-2. The revamped Ectomobile was intended to be called ECTO-2, but was later changed to ECTO-1A. The scene must have been shot before the change was made.
When the Ghostbusters leave the firehouse to respond to the haunting of the crystal store, Egon and Winston wear tan uniforms while Ray and Peter wear dark gray uniforms. At the store, all four Ghosbusters wear dark gray uniforms.
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. Based on her raised arm, the statue alternates between laying facing up and face down between shots.
When the Con Edison Supervisor and the police arrive at the hole on First Avenue, a subway station exit nearly has red lights on top. The only one subway station on First avenue is the L train, and all of the exits to that station have green lights on top.
(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.
Early in the film, Frank tells Dana "I'm not the doorman, Miss Barrett. I'm the building superintendent." At the end, Michael P. Moran is credited as playing "Frank the Doorman." It's a joke, to acknowledge what he said at the beginning.
When the toast pops out while the toaster is dancing, the toast looks like toast, without any sign of slime. He moved the toaster to another location, so he could've put bread in and pushed it down before playing the song.
(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 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.
During the "We're Back" montage, Ecto-1 pulls out of the firehouse with Louis Tully running behind, and piles of wood and metal lean against the back wall. During the rest of the 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. A huge hole was punched in the floor during Ghostbusters , and they most likely had the materials to repair it.
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 shots until the end. The proton packs have activator switches on the wands, to conserve power, so the slime throwers could have the same.
In the statue, Peter stands next to Winston, who stands next to Ray, who stands next to Egon. It the next shot, Winston is next to Peter, who is next to Ray, who is next to Egon. In the following shot, they're back to their first configuration. Considering the space and the amount of time passed, it's entirely possible that they deliberately changed places.
It is established that no one believes in the Ghostbusters or the paranormal. However, in the first movie the citizens of New York were terrorized by ghosts, the Staypufft Marshmallow Man, and supernatural forces in the sky above the skyscraper when Gozer came to Earth. Reporters even filmed the events. Five years later, no one believes anymore.
People still don't believe the Ghostbusters they found slime underground at the trial many days later. In order for the real electricians to get the city lights working again, they would've to go underground to do all the repairs in the location where Ray caused the damage, and they would've discovered the river of slime for themselves.
When the Ghostbusters arrive at the museum in The Statue Of Liberty, they descend from the crown on ropes. The windows on the crown are not large enough for them to fit through on their own, much less with their equipment.
At one point, The Statue of Liberty wades through water up to her chin. The Statue is 111 feet tall from top to toe, and the waters of New York's Upper Bay are a maximum of 45 feet deep. Venkman says "Keep kicking Libby," which means she is swimming, not walking.
Vigo possesses the Museum curator to capture Oscar for him so he can get out of the painting and into the baby's body. The Ghostbusters retrieving the baby when they arrive at the museum, but Vigo is able to exit the painting and retrieve the baby anyway. Vigo is gaining power as the time approaches midnight, so the time combined with being physically close to the baby could've given him enough power to briefly escape the painting to defeat the Ghostbusters.