Factual error: When the barracks are completely flooded, several of the trainees shove shower hoses down the toilets for an "unlimited supply of air" (as Merlin explains moments later, describing it as "simple physics"). However, that old fireman's trick is a technique for surviving dense smoke, which is not under tons of pressure. In the barracks flooded 9 feet deep with water, the pressure would be so enormous that it would easily gush straight down the toilets in a powerful torrent, eliminating any "unlimited air supply."
Factual error: When the barracks are completely flooded, Eggsy punches through a large two-way mirror to escape. The fact is that any glass (or plexiglas) thick enough to withstand many tons of water pressure without bursting would be as impenetrable as concrete to Eggsy's bare fist. He would need a chisel-tipped jackhammer to penetrate such a mirror.
Continuity mistake: When Eggsy is in the interview room at the police station, in a wide shot the tape recorder is on the table by itself, with nothing around it. Eggsy demands a phonecall, the police officer leaves, and Eggsy then reaches beside the recorder and picks up a cordless phone which has appeared from nowhere. (00:16:45)
Continuity mistake: When Eggsy and Harry are walking toward the Kingsman tailors just before he gets introduced to the new weapons, there is a long shot of them walking down the street and two silver cars are coming down the street behind them. In the next shot the two silver cars that should still be in the background have disappeared. Also, as they approach the Lexus parked in front of the Tailor's, and there is no car beside them, in a second perspective, there is suddenly a black vehicle parked behind the Lexus.
Other mistake: When the Kingsmen are looking on-screen at the dossier of the princess, the attached photo of her is just a screenshot from the earlier scene where she is at the dining table with Valentine and the Swedish prime minister, just before she is imprisoned. Her photo was not taken at that point, and couldn't have been attached to her private file.
Revealing mistake: In the Kentucky church-massacre scene, Harry briefly scuffles with a man in a light blue sweater. Harry elbows him in the ribs, sticks a grenade in the man's pocket, spins him around and delivers a crushing blow to his neck, then hurls him head-first into a wall. Throughout this sequence, the man is obviously a live human being. However, a fraction of a second before he impacts the wall, he transforms into a life-size rubber dummy before our eyes, in mid-frame, without any obvious camera cut. This demonstrates the complexity of the scene - it is a digitally-edited collage of action sequences almost seamlessly strung together into what appears to be one long traveling shot.
Character mistake: Anyone who knows anything about firearms would never handle a firearm with their finger on the trigger unless ready to shoot. In the training scene with the dog Eggsy is running with his finger on the trigger the whole time. Merlin also does it in the end scene at Valentine's secret lair inside the jet.
Visible crew/equipment: When Eggsy is in the car yelling at his mum's boyfriend and the windows start to go up, you can see a man's head next him on the floor.
Continuity mistake: When Harry and Eggsy are talking about the newspaper headlines, in Harry's study, once Eggsy sits down, the desk lamp on Harry's desk rotates between shots, the main boom arm also changes angles from vertical to set on an angle.