Plot hole: The whole concept of a kidnapper 'pretending' his victim didn't get on the plane in the first place is utterly laughable. Even if nobody saw her in her seat, why didn't they see her walking down the aisle to her seat, or waiting to board the plane, or in the boarding tunnel itself? Why isn't she listed on the passenger manifest, or on her mother's travel documents or airline ticket? And there would have been security cameras all over the airport.
Factual error: Jodie Foster states that as a propulsion engineer, she knows the internal structure of the plane by heart and has worked extensively on it. The problem with that is that propulsion engineers work on the jet engines - which are all on the outside of the plane, on the wings. Sure, a propulsion engineer might have some basic knowledge of the plane's interior design and engineering, but the in-depth knowledge she claims comes from her job simply isn't right - that's the job of avionics engineers.
Plot hole: Gene's plot has far too many holes for it to have had any chance of working in real life. He somehow must get himself assigned to that particular flight (OK, maybe as a flight marshal he could persuade his superiors to assign him to it) - but also hope that the air stewardess Stephanie is also assigned to that flight - something he has no control over. He then needs to push Jodie Foster's husband off a building, hope the corpse is taken to the only morgue in the city with a crooked morgue director, hope the coffin is assigned to the flight he is on. On top of that, he needs to get explosives into a coffin, the combination of which he actually says he doesn't know. He also needs to hope no-one notices the child, the child doesn't make a noise while boarding, there are lots of spare seats on this inaugural flight, and that mother and daughter move to them.
Factual error: The Royal Canadian Mounted Police are shown checking baggage in the final scenes at the same time that FBI agents are making arrests. The flight landed in Goose Bay, Newfoundland and Labrador, Canada. According to the Criminal Code of Canada, any crime committed during an international flight that ends on Canadian soil is deemed to have been committed in Canada. That being the case, the RCMP as the national police force has full jurisdiction. There is no way the FBI would be involved at that point.
Plot hole: Bomb squad vehicles are present at the airport when the plane lands. There is no bomb disposal unit in Goose. If one is needed it has to come from two provinces away - travel time by vehicle, in March, going flat out: minimum 20 hours + a 2.5 hr ferry trip. Not to mention that the markings on said vehicles are American: transportation of those vehicles from the US wouldn't have been any easier than from Canada, on the contrary, so again, they wouldn't have had the time to bring them over.
Other mistake: At the very end, Jodie Foster's daughter is drugged and not given any medical attention until hours later when she wakes up. Giving medical attention to a person, especially a child, that is the victim of a crime is easily one of the first things that law enforcement would do. Obviously done to make the final scene more dramatic. (01:28:50 - 01:31:30)
Continuity mistake: Near the end when the Air Marshall is getting off the plane to "get the two agents" there is a shot of him from behind walking off the plane with snow covering most of the ground, but when the shot cuts to a wide shot from the side of him all the snow is gone.
Plot hole: The premise of Carson framing Kyle is utterly ridiculous. Not only the fact that the authorities would never accept a ransom demand without speaking to a supposed hijacker, but Carson has absolutely no evidence to give that Kyle is a hijacker, and yet the FBI somehow all know who she and treat her as a hijacker just from her looking at them from the plane door. The movie seems to act as if an Air Marshal's word alone is enough to convince the authorities that a crime or hijacking is happening aboard.
Character mistake: An air marshal would not give such leeway to someone detained for such disruption. The person would be detained and immediately routed to the nearest airport for investigation or arrest.