Continuity mistake: Hastings buys a large scotch for Rupert Carrington. You can see in the wide shot Hugh Fraser with his right hand raised to give the coins to the bartender. But in the next shot, Hastings' hand is not in frame yet, and he even ends up paying with the left hand instead. (00:30:00)
Continuity mistake: At the hotel, the Count is checking in. Behind him by the revolving doors you can see three people having a conversation, with one being a woman wearing a distinct floral dress and brown stole. He steps away from the concierge and the same woman is having a chat in a different part of the hotel, right part of the frame. (00:03:30)
Continuity mistake: When Miss Lemon announces Mr. Halliday, Poirot has two open letters on his desk, side by side when the camera is positioned behind the desk, which turn into one (probably one on top of the other) when the camera is positioned in front. (00:07:00)
Continuity mistake: Hastings and Poirot are talking about the optimist friend. Once Hastings says "It bucketed down", he's holding the newspaper, and it stays pretty firm. Miss Lemon knocks, and the newspaper is now folding at the upper corner. (00:06:45)