Episode #4.3 - S4-E3
Plot hole: The killer poisons the victim's belonging when they met early in the morning - that means that the victim stayed the whole day with the poison in the pages of his journal he always brings with him. Yet the murderer's plan required the poisoning to happen at a very specific and precise moment: the end of the day after they'd all had eaten together the stew with the non-lethal dose. It is something that they couldn't have planned at all the way it is described, not when they did the poisoning so far in advance.
Episode #4.2 - S4-E2
Plot hole: The police staff in this episode act as if it's morning and they just met; Camille goes fetch her boss at his shack after he barely survived some jogging, and he never asked her about the date she had the previous episode, aka previous night (because of course two murders happen in two days in the little island). But the murder happened in the evening, at least well past 5 PM when it rained. Aside from the fact that the light during the first interview with the suspects is perfect and bright for being dinnertime even in a tropical paradise, what have Humphrey & co. been doing all day, if the day is almost over and they are exactly as we left them at the end of the previous episode?
Episode #3.1 - S3-E1
Plot hole: Humphrey never asked for the background check to be extended to the dead sister, but somehow he happens to have a copy of the shoplifting conviction from years before. Moreover, the proposed switch appears a stellar impossibility, since Sasha sold a multimillionaire company she herself founded and Helen is not her twin, but a sister younger by one year that looks 'a bit' like her; not exactly the kind of anonymous recluse who would never receive any visit, hook up with no former affiliate and make in any way plausible a recycled Christie plotline. (00:47:15)
Episode #2.7 - S2-E7
Plot hole: The professor at the beginning of the episode says;"I want the instruments photographed as back-up", which makes total sense. And in fact we do see pictures of exactly that...except one. The one photo that will prove to be fatal for the murderer is a picture not of the instruments, but of the weather station, which would be absolutely useless as backup for being too distant. But had it been a close-up, the 'shadow' gimmick that solves the case would have not happened.
Episode #2.7 - S2-E7
Plot hole: It is established that there are 4 stations in the N, S, E, W parts of the island and it takes roughly the same amount of time to get to any of them from the central part of the island where the university is. When we see a map of the fictional Saint Marie as DI Richard Poole explains how he is going to look for the camera, the map shows an island with a distance W to E about double the the distance from N to S, making impossible that one could get by car to places with such different distance from the center in equal amounts of time. (00:30:50)
Plot hole: The plot resolution hinges on the fact that the culprit gave the illusion of the door being locked while the lock was already broken. And he did that by jamming under the door a rusty old fork, through the usage of a string. However, this appears really far-fetched to say the least; if the door was obstructed by an object under it, it would have not given way as if the lock was busted. It would have dragged, created a screeching noise and scraping the floor leaving visible marks (which considered it's a locked room mystery, would have been investigated). The person breaking in was Jack, even, who is supposed to be really perceptive and crime-savvy and not the average person.
Plot hole: The way Jack narrates the murder does not make sense; according to him, the killer went first to grab the telephone, then back to the garden to kill the victim, then from there, he had to go, unnoticed and with his robe loose, back to the shack (which seems to be close to the entrance and the box where the phone was). Moreover, to stick the rope into the clay sculpture, especially the way we see it, not entwined into a ball but unfurled, he needed to disrupt the sculpture in a way that would have been noticeable, even if the clay was not entirely dry yet.
Plot hole: One of the guests is a journalist going undercover. The police finds him out because...they google his name, "Bryn Williams journalist" (no quotation marks), but then not finding anything (and Jack says so despite actual results being visualized, but he dismisses them at a glance) Florence has a stroke of genius and says "William Bryn then." And this time the googling pays off, with a search result page that says, literally, that he's an investigative journalist "known for going undercover to investigate", apparently being a master at that having won prizes! In all this amazing silliness, it appears quite impossible that they wouldn't know his real name, since they already identified the suspects and ran background checks, which in every episode always include checking with immigration when they entered Saint Marie. He couldn't have entered the country under a fake name.
Plot hole: The murder as described makes kinda sense (it's a device used in other works of fiction), but not at all as shown; the person hiding in the bathroom closes the door to the point of making it click, and opens it also with a click. Even panicked, the other person in the room is going to hear the sound of a lock right behind him. The door should have been closed only partially. Moreover, the bathroom door is right by the front door closed just by the door chain, and Philip Marston had the door unlocked and with just the chain to break; the bathroom door was fully in his view the whole time.
Plot hole: There is simply no way that the cheating presented in the episode, with the player reaching in his pocket to mark the cards, consistently for over 5 years (!), and wearing the same obvious prop, could go unnoticed, especially in official tournaments with millionaire prizes, on camera even.
The Healer - S7-E4
Plot hole: If the person pretending to be Steadman wrote to his fiancée breaking off the engagement, wouldn't she recognise that it wasn't his handwriting? They had known each other for years, surely she had seen his handwriting before.
Plot hole: The author uses a typewriter and avoids technology. His fake identity was built on a laptop kept on another island. But in the big reveal, her guilt is predicated on her having seen the laptop to find out he was leaving her.
Plot hole: The big locked room mystery is such only because the detectives build it up as one; they don't make any remark about the fact that the door can be locked from inside without the need of a key, they don't check the door frame or the lock (which would have revealed that it was bashed open while unlocked) and they do not even consider for a moment the idea that someone could have made a copy of the keys, which is the first thing anyone would have assumed. Not just that; nobody on the floor who has been working at the bank for years says anything about some cleaning lady they haven't ever seen before and that happens to be the first respondant to the murder.
Plot hole: At the beginning of the episode it is established that Samuel Palmer "was putting out his rubbish just after midnight", but in the rest of the case (coroner report, case discussion at the station, question to the suspects) the time of death is 10 PM. The part of the dialogue at the beginning (Dwayne talking to Jack and Florence outside the house) is cut in home video releases, but still appears in Jack's case-solving flashback reel. Other references to the wrong midnight time appear in Cordell' statement as he turns himself in (says he entered from the window " Just before midnight.") and Jack's usual schtick with the suspects ("Shortly after midnight, Eugene's neighbour, Samuel, he heard arguing").
Answer: Most likely they had only tropical exotic drinks, he wanted an old fashioned English beer.