Plot hole: When Jessica is arrested for the murder of the enemy agent. Her purse is confiscated. It was reported stolen by Michael Haggerty. But throughout the whole show none of the cops looked at her ID. Besides that she's a famous women and someone at the police station would definitely know who she is.
Revealing mistake: When the first scene at Emma's hideaway was shown, obviously the two characters Emma and Jessica were filmed seperately on the couch and then the film was put together (as both are played by Angela Lansbury). However Jessica did a lot of moving while she was talking, so Emma was looking straight across at where she had been, despite her bending down while she talked.
Audio problem: When singing "Goodbye little yellow bird" Emma blows a kiss to Oliver offstage, but she is still singing the song when she is blowing the kiss. Although it's possible she was pretending as it was part of the act, her voice wasn't even muffled by her hand.
Continuity mistake: When Emma was almost hit by a car she jumped up onto the edge of the wall and then back down onto the floor. When she landed she was facing the theatre door, but when Oliver came of the door and the shot changed, she was facing away from the wall. This happened in the transition from stunt person to Angela.
Factual error: When Joellen removes the manuscript from the locker at the Yonkers bus station, you can see the address it was originally mailed too. The zip code is 10205. This is incorrect. All Buffalo, NY zip codes start with "14xxx" a zip code starting with "10xxx" would be much further down state.
Continuity mistake: Mr. Giles holds the telephone as he interrogates the private eye, who answers in a rather defiant way to the threat of "the local constable" getting involved. So Giles puts the receiver down, but notice that in the first shot he is passing it from the right to the left hand. In the close-up it is the right hand to hang up, though. (00:23:15)
Continuity mistake: During the scene at the train station, Jessica's nephew introduces to her Kitt Donovan from the publishing company. Just as he says "in public relations", a man pops in the foreground - it is an extra that walks out of the rail car. In the reverse shot that follows, the same extra is only then coming out of the train. (00:09:35)
Continuity mistake: Episode 1-0, "The Murder of Sherlock Holmes"; Mr. Giles opens the door of the antechamber of his office, finally putting an end to Jessica's long wait. Grady right away at the first click of the lock takes his hands off his pockets and tidies up his sleeves. But in the next shot when Grady actually enters and talks to the secretary, Grady is still with his hands on his pockets repeating the gesture. (00:10:10)
Continuity mistake: When the character played by Ron Moody runs into the sea in pursuit of a suspect trying to escape, he shouts at Jessica to "Get the package!" In the next shot we see Jessica reluctantly follow him in, with a clear waterline from a previous take showing on her dark overcoat/cloak.