Katherine tells Emerson the code is 7463. The code that Emerson mouthed, over background music into the microphone, was 7543.
Early in the movie Emerson rips Katherine's jeans to pull out the metal embedded in her leg then ties a gauze bandage around it, yet starting right after that she is running around the station and her jeans are completely intact. After she suffers a gunshot wound to the abdomen, same thing - he carries her around and there is no visible sign of what was previously a grievous wound. In this case its slightly possible all the bandaging is under her sweater but even that is a stretch.
In the late scene featuring a BMW 7 series, the registration number is AK03HXT, when John Cusack drives away in it it suddenly becomes T96 ABT, which is a BMW that is currently legally on the road in the UK.
The dossiers' birth dates and ages are inconsistent. The first few indicate the movie happens between June 2011 and May 2012, while the last few can only be correct between November 2012 and June 2013.
In the credits, Liam Cunningham's character is named "Grey". During the scene where Katherine and Emerson scroll through names on the laptop, his last name is spelled "Gray".
Emerson incorrectly refers to ciphers when describing one time pads - they are in fact codes, which makes it impossible to crack them systematically. They ensure privacy to those individuals who have copies of the pad. It would be poor security practice to derive a pad from a single key.
When the "broadcasters" speak their numbers, they talk into the top of a condenser microphone. These mics pick up sound from the side and broadcasters would know this.
In the Personnel Data Report for Danny Andreotti, the map shows a location in Louisianna (Monroe, LA); this is a typo, it should be Louisiana.
Normally, when a patient is admitted into a British hospital with a gunshot wound, the police are immediately notified. If the patient is/remains unconscious, they are placed under 24/7 (sometimes armed) guard, until they regain consciousness, and can be interviewed by CID/SB/SI/SO19. It is more than likely that Grey, upon learning the situation, either canceled the police or dismissed then in order to avoid any questioning. In the beginning of the film he alludes to his powers in this respect when he talks about having to deal with the police over the shooting of the girl.
In the hospital Emerson tosses his boss a flash drive stick with all the downloaded details of the dossiers. This is solely because in this hospital Emerson was not placed into a patient's gown but allowed to remain in his blood-soaked clothing following treatment.
The train that Katherine uses is a British Rail Class 101 diesel multiple unit. These trains were introduced in the 1950s and the very last of them were withdrawn from public use in 2003, nine years before this film is set. The train is also painted in a British Rail livery not used on public trains since the mid 1970s.
When Katherine is in the hospital, you can plainly see her neck veins pulsing, but the heart rate monitor is beeping away at a completely different rate.
When Emerson and Katherine find David's body in the basement, the shadow of a boom mic is plainly visible on the back wall.
When Katherine begins transmitting the cancel codes, she says "oh" instead of "zero". At the end of the same sequence, she correctly says zero. While some "numbers" stations did broadcast letters instead, they never mixed letters and numbers within the same broadcast (or station). This is likely a mistake by the character due to her state of mild panic in using the colloquial "oh" instead of the proper "zero."
Katherine, the broadcaster, is described as a cryptology specialist, and one of the very few people who can do what she does. Presumably meaning she is very good. Later in the movie she shows Emerson some numbers and says "What does it say?", then looks totally baffled while Emerson explains he can't read it, how a one time pad works, and how the broadcast numbers are used. A one time pad is ultra-basic Intro to Cryptography material. It is totally inconceivable cryptology specialist doesn't know this stuff.
(at around 59 mins) When Emerson tells Katherine to lock herself in the security booth, he claims it is the only secure room left. This is not true since the broadcast room, which has not been breached, is also secure and is locked from the inside.