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.
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.
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: 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).
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: 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: 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.
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.
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 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: 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").
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.
Answer: Most likely they had only tropical exotic drinks, he wanted an old fashioned English beer.