When Emma enters Simon's English country house she leaves the door wide open and continues into the room. In a subsequent shot the door is closed.
The young Simon Templar is obviously left handed but the older Simon Templar is right handed.
While fleeing Tretiak's henchmen 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 and Simon meet in England (after Moscow), there is a flipped shot where Simon's mole and Emma's scar swap sides on their faces.
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 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 the names of Catholic saints." In the case of "Louie Guanella", this is incorrect. Luigi Guanella was a Catholic priest from Northern Italy (b. 1842, d. 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 announcement takes place in the 1st and 2nd year undergraduate teaching laboratory rather than a lecture hall, most likely because it looks more "science-y". No lectures - on even the most mundane subject - would be given in this room, never mind something of this importance.
When Simon raises the heavy iron manhole cover to peek out of the sewer, the cover would be too high for a vehicle to drive over, yet it does, and Emma and Simon somehow have enough room to crawl out from under the vehicle as it is parked over the manhole cover. This would not be possible as the vehicle's undercarriage is only inches from the ground and crawling out then lying underneath the vehicle would require a much more significant clearance.
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.
When Ilya Tretiak (Ivan Tretiak's son) holds up Simon while he is stealing the microchip in the beginning of the movie, looking at the front of the gun, the revolver isn't loaded.
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.
The news article given to Simon Templar in the airport lounge is obviously just a headline over a random article gleaned from newspapers from the time. "Dr Russell wins over sceptics" is over an article regarding Harry Handlesman, founder of the Manhatten Loft Corporation.
When Simon is drinking a glass of wine, the glass is almost empty. When he takes another drink just moments later, the glass is near full.
At the Beginning of the movie Simon's laptop shows he has 42 million in his account. He ask Tritiac for 3 million then later doubles it to 6 million. Yet his account goes to 50 million when it should have been only 48 million.
49 if you count the 1 million for the Micro chip. So how did he hit 50.
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.
When Templar is using a blowtorch in an attempt to cut trough the barrier of solid iron and escape the sewer, he doesn't put the blue (hottest) part of flame to the iron grate; instead he puts the yellow (coolest) part of the flame at the surface. This mistake would require hours more work.