Episode #4.8 - S4-E8
Plot hole: For things to go the way Goodman describes them, Dwayne and JP must have not heard in the deserted courtroom building the noise of the glass of the fire alarm button being shattered in the corridor just outside their position. It's an impossibility. (00:18:00)
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 #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.
Plot hole: When we see the murder happen in flashback, the killer stabs the victim wearing no gloves whatsoever, nor wipes the handle. In a very sportsman conduct, the victim also decides not to literally scream bloody murder as he gets stabbed, nor make any sort of sudden, noise inducing movement that would have instantly exposed what was happening. He gives their killer, apparently, all the time to go back to their accomplice before Catherine notices the blood pouring on the floor - how rude of him to silently bleed all over the booth without cleaning after himself. (00:41:45)
Plot hole: The murder happens where and when it happens because the candidate "is a very busy man", and apparently then the best course of action to kill him is doing it while he is casting his vote. At this operation, involving the other 2 candidates for the role, there is no press nor any normal voter, for no reason - not safety since Dwayne was not expected. Had they introduced the rich Victor Pearce as some sort of mobster surrounded by bodyguards, it would have been an acceptable plot idea, but the guy travels with his son as sole member of the staff and nothing about his characterization leads the viewer to believe that the only chance to murder him is for a rotund 62 years old lady in clogs to perform a Metal Gear stunt sneaking in undetected while a priest is facing the other way for 5 seconds and pray that nobody else shows up at the voting booth and all the others are taking their time to put a cross on a piece of paper.
In the Footsteps of a Killer - S6-E7
Plot hole: Somehow, Jack is so lax in his investigation that he does not ask any detail about the sales representative supposedly the husband had an affair with and that he ended (which could have very well been a suspect nobody considered before, since he ended the relationship to stay with his wife...had she existed, but Jack can't know that he's lying!), but has acquired DNA from Ian Matlock to run an overnight test on the hair sample (which we have to assume was complete with follicles and still in test shape after 8 years in a bag).
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.
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.
Plot hole: Why would Sian let Griff into the shower room if she broke up with him the night before?
In the Footsteps of a Killer - S6-E7
Plot hole: Spoiler. The killer needs to get rid of two women. One is his mistress that he just dumped and is acting psychotic about it, the other is his wife. He kills the wife and makes it look like it was his mistress to murder her. So she goes to jail, where she will die two years later due to pneumonia...and in all this time, she never once says anything about the affair! He has been extraordinarily lucky, because had she said anything, and she had absolutely no reason not to in her circumstances, the case would have not been so open-and-close, they would have considered the idea that he could have been an accomplice, but even assuming the past detective (which DI Richard Poole called a good detective) was a total fool, at the very least Jack and his team would have found a trace of this controversial alleged affair in the files and solved the case much earlier. But no, the plan was to send his mistress to jail and that somehow made her cease to be a threat to him, when instead she'd have been much more dangerous to him.
In the Footsteps of a Killer - S6-E7
Plot hole: The original murder went unnoticed because of extraordinary incompetence of the police; even if they did not have the ingenuity to extrapolate the background noise, they still had to investigate how a murderer would have gotten away with a corpse in the middle of a carnival, but the issue is never raised. More importantly, the last phone call of the victim came from a place 40 minutes away from the city, which would mean a different cell tower - and since they needed to track down the body that never turned out, monitoring the phone the last call came from is standard procedure.
Plot hole: The big climax of the episode involves Humphrey arriving at 1 PM to escort Martha to the airport as he promised, finding the shack empty with a parting letter from her and rushing on Dwayne's motorbike to catch her plane, failing to do so. All very cinematic, but that means that had Martha not left early, she could have not caught the plane at all.
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.
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.
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.
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.
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.
Answer: Most likely they had only tropical exotic drinks, he wanted an old fashioned English beer.