Episode #1.1 - S1-E1
Plot hole: The real culprit pinned the crime on DI Charlie Hulme, but since everything was improvised and acted on the spot, it doesn't explain how could they access Charlie's safety deposit box at the bank stuffing it with incriminating evidence. It is specified through Fidel and Dwayne's dialogue that the box is at the bank, where everyone is required to show identification to get a box and access it, not some anonymous private storage company or a random locker.
Episode #1.5 - S1-E5
Plot hole: Leon Hamilton is the most hated man in Sainte Marie, with over 900 people on the island conned by him. Yet exactly one person in the whole island knows how he looks like, every newspaper and website never ever published a picture of him even during the very public trial when they were trying to get their thousands of dollars back nor publish an archive picture after his death.
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)
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 #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 #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 #4.3 - S4-E3
Plot hole: Five people in total are on a remote location, but they have food poisoning, so severe that they call in the paramedics. Humphrey arrives when they are being taken care of already. During all the time it took for the doctors to arrive, and when the actual rescue was there and was administering help, nobody bothered to check on the victim in his tent - if anyone did, they would have found him dead, since the poison killed him in 30 minutes. The son, the lover and the good friend of the victim, nobody thought about him for a moment.
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 #4.6 - S4-E6
Plot hole: The killer is able to dupe the victim into hiding at the back of a car promising that they will bring a certain someone on a bench and get them to talk about a very bad thing they've been doing, so the designated victim can record the conversation with their phone. This requires the victim to be outrageously stupid; the car, with a closed trunk the victim is hidden in, is parked at distance from the bench; the microphone of her phone would never record that far, and nobody would believe their own phone can work that way, especially when they can hide the phone in a sports bag by the bench, the bushes, the gaps in the bricks or just demand that their supposed friend carries the phone herself. Also, the killer couldn't be sure that the device wouldn't say something about what they were doing (as often happens before you start a recording; you state the time, place, purpose of the recording, plus all the other content of the phone she didn't have time and chance to review!), but makes no attempt to make the phone disappear.
Episode #4.8 - S4-E8
Plot hole: The way the murder happened (where was the gun shot from?) is entirely unknown for the first couple of days, not even a hint - it seems the investigators don't care at all about it, which is quite silly per se. More importantly though, knowing so little about the way the shooting happened would have at the very least prompted an analysis of the shirt to check the entry wound and any residual. During that sort of test the 'other substance' used in the trick would have been easily discovered (visually it is radically different up close and drying out), exposing the culprit.
Plot hole: The whole plan for the murderer to create a perfect alibi hinges on an incredibly precise timing of the victim's action and bodily reaction to the poison, both out of his control. The Governor needed to shake a lot of hands and deliver a speech without yet dropping dead, and everyone needed to ignore entirely the signs that she was feeling unwell. In fact, she needed to collapse as she was drinking, dropping the glass and doing it somewhere where he could squirt some more of that poison in the glass. He couldn't predict the whole situation with the glasses and the Commissioner that would create the alibi (he leaves the party when refreshments are just being served), but for this unpredictable chance to get an alibi he took the huge risk to carry the whole bottle of poison with him, which the plot never explains why was never found by the police or disposed of.
Plot hole: For the plot to go the way it was described, Newton had to wake up at the right time from his drunken stupor (that for some reason the real killer assumed would last eons), sail his boat, go somewhere at sea to dump the corpse, and then back exactly at his pier in the crowded spot where everyone knows him, and do a thorough bleaching of the cabin, all in less than half an hour, close to 15 minutes since he has then to go to Catherine's, which is half a mile away. So around 10 minutes being generous, for a hungover middle-aged man in bad shape and in shock and be there by 7:30, with the real murder happening around 7 (the wife says at 7:45 that it's been 45 minutes since they heard from him). Of course he also had to do that unnoticed, with no record or witness about a boat sailing in and out within a few minutes at dinner time, and likewise nobody saw the real killer running like mad at the docks - also of course, the whole stabbing was done without a single trace of blood getting on him.
Plot hole: Saint Marie is an island with an astonishing homicide rate, but even considering that, with an a missing person and a phone call reporting a stabbing at the boat of a notorious belligerent drunk, with the same person called by name as the culprit, it's quite inconceivable that in a relatively small town like Honore nobody looked for Newton Farrell and questioned him all night - especially being so easy to find, or even checked the boat. Instead, they are surprised by the bleaching of the deck the day after, proving they did not even look for a moment during the night.
Plot hole: Why would Sian let Griff into the shower room if she broke up with him the night before?
Plot hole: The shower room door has a lock, that as shown in the denouement the killer never closed - making it fairly obvious that the victim did not lock herself inside the room to kill herself. Another minor but still significant problem with the stratagem used; the killer shot the girl in the chest; even if they didn't do any gunshot residual test on her hand, the trajectory of the bullet still would have proven the impossibility of a suicide.
The Secret of the Flame Tree - S6-E2
Plot hole: For the murder to happen as described, the victim must have set an appointment at the cliff with a mentally ill, barely coherent person who spends her life holed up in a bungalow and is afraid of everyone, trusting her to remember and be on time - instead of knocking at her door and walk her to the location, that is very close. It's a complete absurdity.
Answer: Most likely they had only tropical exotic drinks, he wanted an old fashioned English beer.