Factual error: In the scene at the World Expo, the flags are being displayed. Next to the US flag is the Bahamian flag. The Bahamas didn't have a flag until 1973 when it gained its independence. It used the British Flag or the "Union Jack" since it was still a part of Britain. Also, the flag disappears a few seconds later.
Other mistake: When Schmidt sets the self-destruct timer at the Hydra base, the clocks count down at twice the speed of real-time.
Continuity mistake: When Agent Carter shoots Rogers while he is holding his shield, she picks up a M1911A1 pistol. When she does, it has the hammer down, but it cuts straight from her reaching for it to her shooting it, without her cocking the hammer first, which needs to be done since the pistol is single action only.
Suggested correction: The Colt M1911A1 is a single-action semi-automatic pistol, meaning when the pistol has a bullet in it's chamber by cycling the slide you don't need to manually pull the hammer down to fire. It doesn't work like a single action revolver where you have to pull down the hammer manually each time to fire. You could, for a shorter trigger, but you don't have to because of the slide. In other words, the pistol Agent Carter picked up was loaded with a bullet in the chamber and ready to fire without having to manually pull down the hammer.
The correction is incorrect. A 1911 pistol is hammer fired, meaning the hammer must fall on the firing pin in order for it to fire. The hammer is down in the scene, which meant the pistol would not have fired. However, since the hammer not being down is only visible when using slow motion (the hammer is obscured by Agent Carter's hand in most of the shot), the original mistake is invalid.
Continuity mistake: During the taxi chase scene, the license plate RY*281F on the cab starts out attached above the single taillight on the left of the vehicle. After Peggy shoots the windshield and she is tackled by Steve, the license plate switches to the opposite side of the vehicle as do the gas cap and a single bullet hole in the rear window. There are several more side swaps like this before the cab crashes.
Continuity mistake: After Steve rescues the 107 division and talks with Peggy in the field the space between them changes from shot to shot.
Continuity mistake: When Colonel Phillips is talking to the soldiers for very first time on the field and notices Steve is smaller then the others, the sun changes between shots.
Continuity mistake: In the scene where Dr. Erskine and Steve Rogers are about to share a drink in the barracks, Dr. Erskine's hand jumps from the bottle's neck to the body and back.