Continuity mistake: During the chase scene involving Rogers and the Hydra spy that happened after Rogers got his new powers, the windshield of the cab the agent was driving was shot twice. In some shots, especially once Rogers landed on the roof of the cab, the windshield has no holes.
Deliberate mistake: In the scene where they break into the train from its roof, you can notice that despite the high speed and the consistent wind, Captain America's fellows have no difficulties in running, standing up still, and even jumping, like they felt no air resistance. Moreover, you can see that their hair is not moving at all, completely missing to realize any "windy" effect.
Continuity mistake: When Captain America is ready to jump out the plane, he is looking at Peggy Carter when he reaches for his goggles, the angle changes to the outside and Cap is seen with one hand out the plane and the other in, with his goggles up. it switches back inside and his goggles are on.
Factual error: The German Stielgranaten - as well as other traditional German WW II hand grenades - did not have a ring and pin like the ones the Allies used, but had a string which had to be pulled hard in order to arm them. Thus they wouldn't make such a distinct 'ping' as they do in the movie.
Continuity mistake: During the final battle on board the 'flying wing' between Captain America and Red Skull, the finish on CA's shield appears damaged and scorched. Yet in the opening sequence of the movie when his shield is discovered under ice inside the frozen 'flying wing' it appears undamaged.
Factual error: The early scene in the movie theatre is in 1943. In the opening newsreel a Sherman Firefly tank is shown. This tank was not made until early 1944 [the long barrel and spherical muzzle brake are distinctive]. Even if we were to assume the tank was made earlier in the Marvel Cinematic Universe due to people like Stark working for the Allies it still would not be shown in a newsreel anyway for security reasons, with German spies and sympathizers present in the US during the war.
Continuity mistake: Colonel Phillips walks into the cell that holds Dr. Zola, while carrying a dinner tray. When he puts it on the table the salt and pepper shakers and the glass of milk are on Colonel Phillips' side of the table. In the next shot you can see Colonel Phillips turning the tray around to Dr. Zola. After he refuses the food and Colonel Philips makes a snappy remark about cyanide, Colonel Philips turns the tray back, so the shakers are on his side. But then when he starts eating they are on Dr. Zola's side again. He does move the glass of milk with his right hand and grabs the knife and fork with his left, making it impossible for him to have moved the shakers.
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.