When Carter is fighting the assassin in her apartment she is thrown against the cooker. As she heads towards it, two pans are seen on the far burners. When she hits the stove the pans are missing, with no sound of them having been dislodged. As the fight continues she picks up one of the pots, which has re-appeared on the stove and hits the assassin with it.
When the alarm in the office goes off, there are clocks visible around the alarm. The time on the New York clock is 10:40. About 15 seconds later, as the agents go into the meeting room, the New York clock has gone back 5 minutes to 10:35. It is definitely the same clock because the alarm is visible next to it.
When the office receives a code red alert, there is a close-up shot of the alarm, where there are two clocks next to it. One of the clocks represents New York time. The other represents London time. The clock representing New York reads the time as 10:40 and the clock representing London reads the time as 4:40. The time difference between New York and London is 5 hours, not 6 (occasionally 4 due to DST).
When Carter is in the diner at the end of the episode she states she has a fork against the rude customer's brachial artery. It can clearly be seen digging into his ribcage/thorax, the brachial artery is in the arm and shoulder.
Though it is possible Carter is aware of this, and could be deliberately making a nonsense threat to underscore the ignorance of the rude customer, or to play up the threat by the use of medical jargon to ensure he is scared away.
Though it is possible Carter is aware of this, and could be deliberately making a nonsense threat to underscore the ignorance of the rude customer, or to play up the threat by the use of medical jargon to ensure he is scared away.
As is often the case in Hollywood productions, the portrayal of a suppressor (aka a "silencer") on a firearm here is woefully inaccurate. While a suppressor does lower the decibel level of a gunshot somewhat, largely helping to save strain on the ears of the shooter (as well as anyone in the general vicinity), it cannot reduce the sound to a whisper barely audibly from an adjacent room, as seen here.
When Agent Carter is fighting the assassin in her apartment, the end of her hair clearly falls into the lit burner of the stove, yet it remains unburnt.
At the beginning, set in 1946, Peggy makes a cup of tea with a tea bag; although it seems that tea bags were invented in 1908, it would have been unlikely she would have used one in 1946 as they were only commercialized in the early 1950s and only widely used from the 1960s.
Colleen sneezes into her elbow like a 21st-century person when she and Peggy are pulling the Murphy bed back down.