(at around 28 mins) In the Springfield newspaper montage that span decades, the story beneath the headline is always the same (something about an ex-Padre players' divorce).
(at around 19 mins) When Captain America takes down the search light tower with his shield, the shield can be seen still laying in the debris of the tower as the superimposed shield is returning to him.
(at around 19 mins) In the Nazi stronghold, Captain America prepares to throw his shield at the searchlight. The light operator, fearing the worst, ducks down out of view. When the shield knocks over the tower, there is no sign of the guard, giving away the fact that the collapsing tower is just a miniature.
(at around 1h 12 mins) When Steve Rogers and Sharon fall from their bike into the sea, her stunt woman's skirt flies up revealing a bathing suit.
(at around 16 mins) When Steve Rogers/Captain America goes on his first mission, the flag on his uniform has 50 stars. It would have been 48 stars during World War II.
(at around 40 mins) When he thinks the future world is a Nazi trick, Captain describes a movie he saw with a similar plot. He seems to be referring to 36 Hours (1964) which was not released until 22 years after he was frozen.
(at around 6 mins) When the camera zooms into the window to Steve's home in opening scene, the camera is seen in the reflection.
(at around 57 mins) When Steve and Sharon drive into the parking lot in front of the Diner above the secret base, the shadows of the camera crew on the end of a boom can be seen in the carpark.
At 15 minutes, when attacking the German base. Captain America pulls a German truck for cover, you can clearly see the wires that pull the truck forward, giving the illusion Cap is very strong and pulls the truck.
At 1:32:00 when Captain America throws his shield towards Red Skull, you can see wires pulling the actor and the shield through the air.
(at around 1h 3 mins) When Steve and Sharon exit the diner, the parking lot is entirely empty, despite the diner being full of people.
(at around 23 mins) The missile flies from Europe to Washington DC, then all the way to Alaska before crashing. The fuel needed for this trip would have taken up more room than the missile itself.
(at around 1h 15 mins) Despite being able to rip the cover off of a furnace in the fight below the diner, Steve in unable to open the locked door of the little white car when Sharon locks him out.
The Red Skull is taken from his family in 1936 as a child to be experimented on. Seven years later when confronted by Captain America, he is a full grown adult when in actuality, the Red Skull would have been a teenager.
(at around 28 mins) In one of the Springfield hometown newspapers, the name of the town is misspelled "Sprinfield".