The father of Gertrude Aldridge is repeatedly referred to as "Sir Aldridge". If this character was a British national who owned a title of knighthood, then he would properly be referred to as "Sir" followed by his first name, not his surname. E.g., Sir John Gielgud was "Sir John" not "Sir Gielgud".
Patty says the subway tunnel runs beneath the jail where the first electric chair execution occurred. The first such execution (of William Kemmler in 1890) was in Auburn, New York, which is nowhere near NYC.
When you see the second shot of the outside of the Chinese restaurant/Ghostbusters base of operation (right after the subway train encounter), you can see the Ecto in the garage...this is before they even got the car from Patty. However, this mistake was digitally removed for the Blu-ray release.
As the trio celebrate their first supernatural encounter outside the haunted mansion, Erin (coated in slime) hugs Abby and ectoplasm sticks to Abby's clothes. In the next shot as they continue hopping up and down there is no slime on Abby.
The building housing the Ghostbusters office above a Chinese restaurant is shown to be next to a vacant/parking lot. But in one establishing shot from another angle, there is a large building where the parking lot should be.
When Patty character saves Jillian from the flying ghost axe near the end of the film, the 2 characters momentarily swap places with Jillian moving from Patty's left to her right as they fall and then swapping back as they hit the floor.
As they are about to go to the Aldridge mansion Holtzmann is wearing dungarees when she grabs the pack. When they leave the building she is wearing a black leather jacket over the dungarees but we never see her put it on.
In Rebecca Gorin's scene, Holtzmann is wearing a contraption on her head when Patty tells them to come up to the roof, but when they reach the roof the contraption has been replaced by Holtzmann's glasses.
Erin pops the Stay Puft marshmallow balloon with a Swiss army knife. There is no way a hole the size of a knife would have made it explode - it would have just deflated while the gas inside leaked from the hole.
At first glance, it would seem that Erin has awards from various physics organizations on the wall of her office at Columbia University, but that they all use the exact same formatting, including the same font. Upon closer inspection (high-def Blu-ray helps), they are actually different awards from the same organization ("The American Physical Society"). As such, it would not be unreasonable for them to all have the same formatting and fonts.
Erin says "Mercado" is Spanish for table; mercado is Spanish for market. It's running gag that Erin is really bad at Spanish. Earlier, she mistranslated what the theater janitor was saying when he went past on a gurney.
When Abby is being thrown around the alley while testing the proton pack, the position of her hands changes. If she could move her hands enough and still keep hold of it, why didn't she just let go? Her finger wouldn't have been on the trigger any more, so she wouldn't have been in any danger.
Prior to the Ghostbusters' battle against Rowan and his ghost army, the team used their proton packs and other equipment to capture ghosts. However, in the battle, their equipment is inexplicably used to destroy the ghosts.
When possessing Abby it appears that Rowan has access to her memories as he knows the identity of the other Ghostbusters (and after possessing Kevin comments on him knowing how to ride a motorcycle). However in their confrontation later, Rowan claims to have not known Kevin's name until it is mentioned to him. This makes sense, however, because Rowan had been following the Ghostbusters' exploits before his first confrontation with them. Kevin is never shown in any of the quartet's in-movie promotional footage, so Rowan would not know of his relationship to the Ghostbusters.
All damage to property and people caused by either the portal, Rowan or the Ghostbusters themselves, seems to magically repair itself following the final battle. The people outside the hotel in particular would undoubtedly have been killed when Rowan broke his way out, but there is no indication of this on-screen.
When the cops & soldiers wake up after the portal closes there is no sign of Kevin until he appears with a sandwich. How did he wake up before the others and why did he not see Holtzmann & Patty directly in front of him?
After Erin and Abby fall into the inter-dimensional gateway and are pulled back out, the hair on their heads has turned stark white, but their eyelashes and eyebrows are unaffected. (Neither character appears to have been wearing heavy eye makeup.)