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)
Continuity mistake: Dwayne realises that he did not ask the cute girl's name, and looks at her now far away. Something is wrong with this shot of Joséphine Jobert and her derriere; walking down the same street there's the same couple that is having a drink right next to Dwayne (his black shirt with colourful drawings is not exactly hard to miss). (00:20:50)
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.
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.
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).
Other mistake: Torey Martin says that he had an accident 5 months before, but the car crash evaluation is signed 31/07/2016. Not only it was a Sunday - unlikely day for the car shop to be working -, but that would put the events of the episode on January, contradicting the bank statements and what was established in previous episodes.
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: 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.