When the CIA are tracing Ethan Hunt's phone call from London, the CIA agent reports "He's in England" when the only confirmed digits are 44, from which you can only deduce a UK telephone number. You need another 3 or 4 digits to determine an area and hence part of the UK. (However, it's not unusual for people to misunderstand the distinction between England and the complete UK.)
When Ethan is describing the vault security systems, the agent places what is clearly his coffee cup at the edge of the desk. After he leaves the room, the camera pans back, and now it is instead a clear plastic cup filled with some sort of red beverage, with condensation forming on the outside, which then drops onto the floor, setting off the alarm.
Ethan reaches out and catches a drop of sweat falling from his head while two inches from the floor in the CIA Terminal Room. There is not enough distance between him and the floor for his hand to stop the sweat the way it is shown.
At CIA headquarters, the fire alarms in three sections are tripped. Then Luther hacks into the alarm system to trip the alarm in Section 21. Later in the Situation Room, the alarm is displayed in a corner of the wall display, but Section 21 isn't lit up.
Claire places the tracker on the NOC list's technician near the top of his left shoulder. When we see him in the bathroom in the overhead shot, the grey square has dropped to his lower back on the left side, closer to the center. It is visible when he reaches to clean the sink.
When Donloe the computer technician enters the vault after his unfortunate coffee break, we see him opening the door and stepping in onto a white floor - however, the next shot from above shows a shiny black floor, as it should look when countermeasures are off (it's the same shot where Ethan's reflection is missing).
When Ethan first meets Max while tied to a chair with hood over his head, and the hood is removed, a black fiber from the mask is clearly visible on his blue right collar and subsequent shots but it then disappears.
In 1996 the only scheduled London-Paris passenger train via the Channel Tunnel is the Eurostar, not the TGV. Although TGV is also a generic term used in France meaning any "High Speed Train" (Train Grand Vitesse)
When the ticket for the high speed train is produced, not only does it claim wrongly to be for the TGV, which is actually called Eurostar (TGVs only travel within France and into Switzerland), However Eurostar tickets are available which permit onward transit via SNCF TGV's and Vice versa. It also purports to depart from the "London Terminus". There is no such station - at that time, Eurostar left from London Waterloo.
When referring to the hardware necessary to break into CIA database, the Hunt group discusses about a new "686 experimental RISC" hardware. Although the reference could be totally fictional, it does remind of the way Intel used to brand their CPU back then, when 586 (or pentium) processors were the top of the line. The mistake is that the whole series of x86 processors were CISC processors and no RISC processor has ever been produced by Intel in the x86 series in the nineties.
The Eurostar train is an electric train (like most trains in western Europe), which usually runs on 25'000 volt overhead lines. This makes the scene on top of the train, and a helicopter landing on the roof of the train, impossible. Until a proper overhead line was installed on the English side, though, the Eurostar train had a capability to run on a third rail system using 750v, which is common in England.
When Ethan arrives back at the safe house, he steals a light bulb in the hallway and crushes it. The amount of glass he then spreads on the floor (as a warning device) is far more than the bulb would've provided.
The radius of a rail curve for the TGV is very constrained to accommodate the high speed of the train. However, we see several shots where the train is taking one fairly steep curve after another. With its actual speed, it would just go straight and derail immediately.
After Krieger swats the rat and almost drops Ethan to the floor of the vault, he is barely able to hold on to the rope for Ethan to finish the job. Suddenly as the technician comes into the vault, Krieger is able to pull Ethan up at high speed, faster than is humanly possible. It would take a fast moving reel or vehicle to pull up a human at that rate.
Although some would consider a computer room equipped with the three high-tech security systems, meant to prevent any intrusion, as missing the "most basic security measure" of video surveillance, video surveillance is often not permitted in highly classified areas where there is a risk that unauthorized personnel - such as the security personnel - may see the material. The security personnel may theoretically be cleared for that classification but they would not have a "need to know" the information contained within AND video surveillance visible in another area would defy the measures taken to keep the information in a secure room.
After obtaining the NOC list, Ethan Hunt sends a meeting request to Max at the address Job 315. The original Usenet group and email was Job 314. This is not a goof but merely an increment in the passage number to indicate a new job. Indeed, Hunt then quotes from Job 315 at the end of his message to Max.
The actual NOC List on the CIA vault computer is protected only by the pressure sensor, the temperature sensor, and the audio sensor. A computer with confidential information like that saved on it would almost definitely lock or shut off completely (and refuse to reboot) whenever the other counter-intrusive sensors were on. This would prevent an intruder (if they got past the sensors, as Ethan did) from even entering a password, let alone getting any information from the computer.
In the first shot of the TGV, when closing up, the brake indicator of the train shows red, which means the brakes are fully applied. In normal operations, the indicator shows green, which indicates the brakes are fully released.
When in the tunnel, the Eurostar uses an overhead power cable system (a catenery); the TGV uses one at all times. No such power cable, which would have made the helicopter interactions impossible, is ever shown.
In the computer room scene, when the man enters the room with Ethan suspended above him, you can clearly see the room reflected off the floor, as well as the man who enters the room. However we cannot see Ethan, and as the man was looking at the floor when he entered the room he would have noticed him.
At the end of the film when the helicopter is flying through the Channel Tunnel in the UK, a train passes in the opposite direction. The Channel Tunnel uses separate tunnels for each direction of travel. Trains do not pass each other inside the tunnel.
At the end of the movie Ethan and Luther are sat outside of the Anchor Pub in Bankside. Luther takes his glass to make a toast with Ethan and he goes to drink but the glass misses from his hands. Seconds later, the glass appears again suddenly and Luther drinks.
Sarah rushes up to the fence to find Golitsyn being stabbed. But later during the flashbacks, when Golitsyn is being stabbed, Kristin Scott Thomas can be seen waiting in the background adjusting her coat collar before rushing to the fence.
At the end of the movie when Ethan and Luther are sitting having a drink outside the Anchor Pub in Bankside, they are sat at tables located in a place that would not be allowed by the local council because they would be obstructing the right of way. The only outside tables allowed at that pub are those located on the riverside podium - which can also be seen in shot.
Jack is killed by a speeding elevator ramming into the top of the shaft. The elevator is moving at at least twice its normal speed. If an elevator moved in either direction at an abnormal speed like that, the safety device would've reacted immediately and pulled the elevator's brake (it is used to prevent free falls in case the cable breaks). The killer wouldn't be able to turn it off from the computer (because it depends on centrifugal force alone); and probably wouldn't risk disabling it before Jack got on the elevator, for the safety of the regular passengers.