When Emma first climbs into the underground pipe system with Simon, you can see that she is wearing black athletic knee pads, thanks to the visible white label that says "Trace", an athletic gear brand.
When Templar is attempting to cut trough the barrier of solid iron in his escape in the sewer by using a blowtorch, he doesn't put the tip of the blue torch at the surface, he put the middle of the yellow torch at the surface. This method would require hours of work.
The dates on Templar's bank account transactions do not match the dates he gives in his narration. When he transfers the microchip money, the dates on his computer are for March. A few days later, when he is scouting Emma's apartment, he records the date as November.
When Tretiak arrives in front of the US embassy in search of Templar and Russel, his Land Rover 4.0 se has original equipment proper wheels on it. However, after it explodes and lands, we can see that it now has the three spoke wheels from the earlier generation Range Rover County.
When Simon is breaking into the vault at the start of the film, he is shown attaching his safe cracking device to the safe and setting up his video receiver. Shortly after, he is shown watching the video feed, and, again attaching the device and setting up his receiver.
While fleeing "Tretiak's goons" on the rooftop of the apartment building, Simon is wearing a black hat. When he follows Emma through the roof access, the hat has disappeared and his hair is perfectly styled.
When Emma enters the cottage in the English countryside, she steps to the desk where her cards are laid out. On the desk is a 14" NEC CRT monitor, which is all black. Moments later when Simon appears the monitor is now gray with a black fascia.
When Emma is looking through suspect sketches at New Scotland Yard with the two detectives, the female detective says that the suspect is known by many different names, including "Louie" (or "Louis") "Guanella". Emma's response is "Of course... all names of Catholic saints." In the case of "Louie Guanella", this is incorrect. Luigi Guanella was a Catholic priest from Northern Italy, born in 1842 and died in 1912. Guanella was beatified in 1964 but has NOT yet been canonized (made a saint) by the Roman Catholic Church. He is therefore referred to as Blessed Luigi Guanella, not Saint (Luigi) Guanella.
When Dr. Emma Russell gives her lecture at Oxford about her amazing cold fusion breakthrough, the lecture takes place in the 1st and 2nd year undergraduate teaching laboratory most likely because it looks "sciencey." No lectures are given in this room, even the most mundane, never mind something of this importance.
When Dr. Russell is shown the sketches of Simon's saint alter-egos, the nerdy reporter is in the pack. At the end she is surprised to find out he was the nerd, and the detective from Scotland Yard sitting next to Simon does not recognize him in his nerd persona.
Agnes is on the wrong side of the banister when the dogs jump at her. When she falls she is on the same side as Simon, yet when she goes to kiss him she is on the other side. Had she been on that side she wouldn't have been knocked down.