During the escape from the KGB office, Anna is wearing stockings but when she escapes from the archive, she's wearing pantyhose.
When Lenny is waiting for Anna, outside the Embassy in the car, his seating position changes to coincide with camera angles.
During the escape from the KGB office, when she slips over the stair banister, not only are Anna's stockings and suspenders replaced by tights, but the stockings and suspenders are restored shortly afterwards.
When shown viewing Anna at the park table, Alexei's jacket is heavily-worn shiny smooth brown leather with a dull black shoulder. When seen from Anna's point of view (a reversal that happens at least six times in quick succession), he is wearing a shiny black leather jacket with heavy grain.
When Anna goes to the apartment provided for her by the KGB, she has a wheeled cabin bag with telescopic handle. This type of bag would not have been available in the era of the scene. Actually, two-wheeled luggage was available in the 1980s and early 1990s, but pieces were very rare and very expensive. It does make perfect sense for a high-level fashion model to have one.
During Annas first photoshoot multiple film rolls are shown piling up on a table. However none of them was exposed. An unexposed roll can be distinguished from an exposed roll by its visible paper sleeve (which is yellow for the shown Kodak Portra film) and and backing paper print. When the film is exposed, it is transferred to another spool in the camera and will be reversed. Later it will be sealed with a usually white paper strip that can be labeled with a pen.
When Anna retrieves the CCTV recordings from the hotel security system she takes a hard drive. CCTV systems of that time would have recorded onto video tape.
There are numerous instances where the technology on display (laptops, high resolution displays, webcams, hard drives, flash drives, internet, etc) exists far earlier than the year it was actually released/available.
No ATM or mobile phones existed in the Soviet Union during the '80s. The technology used in the movie is totally out of tune with the time setting.
Agent Miller tells Anna that he could have her shipped to the Czech Republic, but the year is 1990, the Czech Republic did not exist yet, Czechoslovakia didn't dissolve until 1993.
The city of Saint Petersburg is mentioned but the year is 1990, the city would have still been known as Leningrad, they didn't rename it to Saint Petersberg until 1991.
When Anna delivers the phone to Olga, a camera plus operator is reflected in a big polished brass plate on the door.
It is impossible to make a copy of the hard drive as depicted in the film, if only for the fact that the hard drive has no power attached.
When Anna is murdering people for the KGB, she also kills a lot of innocent people instead of using wit/charm/better disguises etc. This alone contradicts the premise of being inconspicuous. Even more, the amount of bodies all around Europe would have sooner or later raised a lot of questions/inquiries by various police forces. With her face known everywhere due to her cover job as a model, the probability of her being recognized near a hit increases dramatically with every assignment.
The film plays with its time line frequently, jumping around between years. Some events ( Anna's recruitment, training, modeling career, and espionage work) seem too compressed to realistically occur over the time frame suggested, particularly in relation to Cold War and the dissolution of the Soviet Union.
Cillian Murphy's character carries a Colt 1911 handgun as his duty weapon. This weapon is a single-action only handgun which requires the hammer to be cocked back before firing. In every scene with him pointing the weapon the hammer is always forward. A trained CIA operative would not make this mistake as the gun would not be able to fire with the hammer in this position.
Upon receiving the gun for her initial assassination, Anna only learns the magazine is empty when she attempts to fire the weapon. A person highly trained in firearms use would likely notice the weight discrepancy and in any case would instinctively check the weapon to confirm it's loaded.