The Adventure of the Clapham Cook - S1-E1
Continuity mistake: Hastings is reading a letter that makes him go "Good Heavens." He is holding the letter by the top left corner, which becomes the top right in the (lack of) reaction shot from his Belgian friend. (00:24:20)
Continuity mistake: When the secretary leans in to console "Henry" after Walton has been escorted away, he pins her handkerchief hand to the desk; in the next shot the hand is turned differently and caught in midair. (00:07:45)
Continuity mistake: During the entirety of the scene in the kitchen, Patricia keeps her hands in her lap, but they change repeatedly from being side to side, to her right hand lying on top of the upturned left. (00:22:40)
Continuity mistake: When Pamela Lyall is talking to Poirot before his scheduled leave going as far as wishing there were a murder so he could stay, her right hand moves on and off the table between shots. (00:24:20)
Other mistake: When Hastings pulls Poirot's slip of paper out of his pocket, he's reading it upside down - as you can see from the close-up, the torn up part should be at the bottom, not at the top. (00:11:40)
Character mistake: Somehow, Poirot is the first person to get to the basement - he obviously would have slowed down terribly the policeman and the young and spry Jimmy rushing down the stairs with him.
Continuity mistake: Mrs Clapperton puts down the score notes and pencil before lighting up her cigarette. Between the shots and the reverse, the pen disappears and the notebook changes position. (00:13:35)
Continuity mistake: When Miss Lemon announces the anonymous caller to Poirot through the window, she holds the receiver at noticeably different heights between the consecutive shots. (00:09:00)
Continuity mistake: When Miss Henderson gets back aboard, the same extra (bearded man with black robes and brown belt) walks by coming from both directions at once. (00:26:30)
Visible crew/equipment: At the very end of the episode, as Poirot's car drives off headed to the harbor, filming equipment is neatly reflected in the door. (00:51:40)
Other mistake: At the very end of the episode, Miss Lemon stops talking as if she could see Poirot glaring at her above the newspaper - while in fact, he is facing the other way.
Continuity mistake: Once Poirot hypothesises that the lightbulb could have been replaced, he curls back his finger pointed at the young fellas. New shot, and he is back pointing fingers. (00:18:45)
Audio problem: In an empty golf course, at such a close distance as shown, there's no way Miss Plenderlieth would not hear Hastings calling Poirot, who "lost another ball." (00:41:20)
Four and Twenty Blackbirds - S1-E4
Continuity mistake: Hastings carries the picture of the two brothers from Dulce to Poirot holding it by the bottom, top, side, depending on the angle. (00:13:45)
Continuity mistake: The story as presented would be impossible. In the story, Poirot tells his audience that the victim was dead at a time which is earlier than the time that he later tells his audience the victim was killed. (00:37:00 - 00:44:00)
Other mistake: This adaptation opens with a nice fake Pathe Gazette news reel. The voice says that "last year" Farley sold a record 5 million pies, and that the new wing of the plant is due to open. At the inauguration, there's a banner behind him saying "1885-1935" and he's saying that in 1935 they sold more pies than at any point in their history, and he gives the kind of speech you give at the beginning of a year, trying to set a new record while still on the hot trail of the previous. At the same time, Miss Lemon tells Hastings that the typewriter broke 'last Easter' and she burst out with Poirot that she has been complaining 'for the last six months', which would put the event at the end of the year.
Plot hole: In the original novel, the victim's voice is described as "shrill." In here, it's quite the opposite. When the trick for the alibi is performed, the relevant lines are not read by John Normington, thus Poirot's exposition at the end, with the girl providing the voiceover, indirectly further exposes its unbelievability.
Four and Twenty Blackbirds - S1-E4
Factual error: Great accuracy went into the dates of this episode - Hastings is following the outcome of the so called "Verity's Match", the 2nd test of the 1934 Ashes series, Australia vs England. The events then should happen between the 22nd and the 25th of June, 1934, compatible with the murder happening on the 16th, and being discovered 3 days after. This however puts it a year before "Murder in the Mews", previous episode where Poirot's dentist was referenced, and that happened in 1935 as stated in the letter to the Chinese laundry.
Revealing mistake: Towards the end of the preliminary murder scene inspection, look at the victim in the last close-up, following the part when Poirot picks up paper from the basket. Specifically, the gun hand: the pinky finger of the supposedly dead person, after being too long in the unnatural position, trembles a little. (00:07:30)