Peggy begins bleeding through her blouse when she ruptures some stitches while manning the radios with Daniel, yet when they are outside later there is no trace of the bloodstain.
After the secret meeting Dottie opens the closet with her right hand but in the next shot she is holding the door with her left hand.
When Peggy & Daniel are talking about his engagement breaking off Peggy's right arm varies position between up in the air to nowhere in sight.
When Peggy first enters the lab (just after the 5 minute mark) she shuts the door behind her. There is even a sound of the door closing and latching. When the camera angle switches, the door is suddenly ajar. When the shot changes again, it is once again closed and remains that way throughout the rest of the scene.
When discussing what sort of weapon he might be allowed to carry, Jarvis asks Peggy about "nunchucks." "Nunchuck" is both singular and plural, thus "nunchucks" is grammatically incorrect.