(at around 1h 19 mins) Konstantine Konali's name is spelled with Ks throughout the movie. In the end credits, his first name is spelled Constantine.
(at around 1h 7 mins) Towards the end of the movie, when plants are used as disguises to approach the "country home", the only round plant crosses the same sidewalk twice.
(at around 1h 4 mins) During the climactic chase scene, the skyline changes from cloudy to clear and back in every other shot.
(at around 16 mins) When Cmndt. Rakov is briefing the Americans, he underlines Konstantine's name. The line changes between shots.
(at around 6 mins) On the classified "Assignment: Moscow" police file, Callahan's name is misspelled as Callaghan.
(at around 15 mins) The Russian flag on the wall of Aleksander Rakov's office is upside down.
The Aeroflot plane shows the Hammer and Sickle on its tail when landing in Russia. The Soviet Union was dissolved in 1991, three years before.
The safe keeps unlocked in any scene it is seen "unlocked". There's black tape covering the latch (which is likely removed as this 'J.I.S.' manufactured safe needs both the correct combination and the key turned clockwise to unlock it).
At 40:35 a metro car is seen. Despite the film having been officially filmed in Moscow, the metro car is clearly from Kyiv, Ukraine (colours blue and yellow, with specific logo on doors)
(at around 1 min) In the beginning of the movie, the male reporter plays "The Game" on a GameBoy console with no power and no cartridge. All subsequent consoles shown have cartridges inserted.
(at around 1h 18 mins) Obvious stunt doubles appear during the end credits. There is no scenery behind the actors as their names and roles are flashed on screen.
(at around 1 min) The male newscaster's GameBoy is in color. Nintendo released color GameBoys in 1998, four years after the movie was released.
During distant shots in the circus scene, the ringmaster's voice can be heard over the loudspeakers, even though he is not seen talking into the microphone, nor is his mouth moving.
"The Game" is played on a Nintendo Game Boy, but the sound effects heard are actually from the Atari version of Donkey Kong (these sound effects are used in many depictions of video games in various TV shows and movies).
(at around 53 mins) When a Russian cop swings on the chandelier to avoid the floor laser grid, hands are clearly holding his legs.
(at around 23 mins) Tackleberry refers to his side-handle baton as a "PK-24". The model of baton is actually the PR-24.