Plot hole: Kyle is an airplane engineer and designer (decent paying job) who is flying on the same plane that she helped designed, and she is traveling with her daughter back home to family after her husband's untimely death, transporting his body in a casket. No one under these circumstances would be traveling lightly (possibly no economy) as she does. She would almost certainly have special privileges and notice from officials, and there would be impending knowledge of her flight prior to boarding.
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: 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: 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.
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)
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.
Factual error: Human remains, while sealed in a casket, are always transported in a protective box. This box, usually cardboard, has "human remains" placards all over it.
Other mistake: When Kyle enters the bathroom to climb up in the room above it to mess with the wires, she locks the door. Yet the door opens for Carson to check it.
Suggested correction: Actually, she escaped so that's why the door is now unlocked.
Suggested correction: Lavatory doors can easily be unlocked/opened by flight crew.
