Factual error: When the helicopter blows up, the train stops dead in its tracks almost instantly. A high-speed train like the TGV needs more then two miles to come to a complete stop from its top speed, or at least from the speed it's traveling at. And a Little Bird attached to it wouldn't make the slightest bit of difference.
Factual error: When Ethan is lowered into the computer room at Langley, he is dropped and nearly lands on the pressure-sensitive floor. When an object moves through the air, it pushes the air in front of the object. As the floor sensors were set so high that a drop of water was sufficient to set an alarm off, the air displaced during Ethan's fall would have triggered them too.
Factual error: The vents that Hunt and his sidekick crawl down at CIA Headquarters are standard galvanized iron box vents; they are very common in the building trade. Try walking or crawling down one - you'll make a noise like the sky is falling down. People will be able to hear you for miles. Every person in that building would know somebody crawling about in the vent system. (This error applies to dozens of films, not only this one).
Factual error: In the scene in the aquarium restaurant where Kitridge reveals to Ethan that the operation was a mole hunt Ethan uses the chewing gum explosive to smash the glass and make his escape. However, the the explosion also causes a man within the aquarium restaurant to go flying through the air and through the window. It is physically impossible for a chemically composed explosive of that size, that is not enclosed, to create enough force to move him in that way, add to the fact that he is seemingly the only person or object (besides the aquarium glass) affected by it, all these combine in making this scene physically impossible.
Factual error: Luther asks for a laptop with an "AI RISC chip". This is nonsense - RISC stands for Reduced Instruction Set Computer and is a description of the number of registers used by the chip - nothing to do with AI, which is a broad range of analytical and non linear heuristic techniques to enable a computer to learn and make non-deterministic (ie. non programmed) decisions.