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.
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: The windscreen on the flying wing appears to be completely intact when it is discovered in the present. Not only is it implausible to believe it survived a head-on crash without being damaged, but also during the final fight on board, it receives several bullet holes.
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.