Factual error: Marie is a female name. Therefore, the island should be Sainte Marie, not Saint.
Other mistake: Inevitably the officers have to concisely give to the quirky inspector a quick profile of the suspect. For their "background checks" they read from profile sheets. It is unclear if they wrote them themselves or not; sometimes especially in earlier seasons they seem to receive them. These sheets contain sentences often written in a rather informal tone that does not fit the format, and in fact often you can see that whatever the officer is saying in casual tone as if it were part of the dialogue is written verbatim on the sheet itself. Example; 6-8, when Florence says "He's big on family values and promoting education in the community", with her line already readable word for word on the sheet in Jack's hand in the previous shot.
Factual error: The fictional island of Saint Marie where the series takes place is a British territory - after all, the whole gig is about fish-out-of-water British cops in the tropics. Specifically the Commissioner explains the history of the island to DI Richard Poole driving him to the station the first time and says that it was a French territory but in the mid 70s they handed it back to the British. Despite that, throughout the whole series the police and almost every local drive (on the left) vehicles with French registration plates, specifically with the 971 code of Guadeloupe, where the series is shot. In the first seasons the British police also routinely sends evidence and other critical parts of the job to labs in Guadaloupe, effectively out of the country. Seems quite absurd that in 40 years the British wouldn't have established vital parts of administration and rely entirely on French neighbours.
Revealing mistake: In most episodes of the season, the computers at the station lack the power cable. It might make sense in the very first episode when it is mentioned that they aren't in a working state, but under Richard's guidance they routinely use it afterwards. For instance in episode 3 Camille during Tom Hilton's interview, Camille is looking at an active computer screen despite it being unplugged.
Episode #1.1 - S1-E1
Continuity mistake: In the opening, two men (one turning out to be future longest serving original cast member, Danny John-Jules) are playing one-to-one basketball. A couple guys are mending nets just by the hoop...but are gone right away in the faraway shot. (00:00:15)
Episode #1.1 - S1-E1
Continuity mistake: Sgt. Lily Thomson drives detective inspector Poole to the beach shack. In the opening wide shot she has her left hand on the steering wheel, but in the closer angles the hand at 12 o'clock is the right hand instead. (00:09:10)
Episode #1.1 - S1-E1
Continuity mistake: Following James Lavender to the beach, Richard calls him "a little proprietorial", DS Lily Thomson between cuts has alternately her right hand on her hip or not. (00:14:00)
Episode #1.1 - S1-E1
Continuity mistake: The 'cleaning lady' hands Richard a used paper tissue. In the close-up of her hand he is picking it up by a messy, crumpled part, but in the next shot Ben Miller is holding it by a side pretty much straight. (00:19:25)
Episode #1.1 - S1-E1
Continuity mistake: DI Richard Poole is in front of the panic room, explaining how the murder happened. He is holding two evidence bags; the fact that the one with the .22 gun changes position between shots several times is made more obvious by the fact that the tag is visible, and each time Richard's fingers are covering different letters. (00:44:20)
Episode #1.1 - S1-E1
Continuity mistake: Richard just closed the shack leaving the pet lizard behind, and is talking to the Commissioner who roped him into staying. Fidel and Dwayne are looking at them; the lighting in close-up differs noticeably from the wider angle (check out of the shadow on Dwayne's arm and on the wall). (00:50:35)
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 #1.2 - S1-E2
Continuity mistake: When coming out of the sea at the start, the guy's hair goes from sleek to messy and back between shots. (00:02:00)
Episode #1.2 - S1-E2
Continuity mistake: Richard is walking through the market fending off the various street vendors. There's a POV shot ending with a woman wearing a headscarf that offers him something that "looks absolutely delicious." In the following shot Richard is still dodging the woman, who gives way to him moving aside in an entirely different way. (00:02:05)
Episode #1.2 - S1-E2
Continuity mistake: When Richard is walking in the market a woman holds out an item for him. The camera changes and this item is now held in a completely different way. (00:02:30)
Episode #1.2 - S1-E2
Other mistake: Camille sits down and orders a large cocktail with a list of what she wants in it. This is then brought to her within 20 seconds. Not possible given the distance from the bar and the order. (00:04:00)
Episode #1.2 - S1-E2
Continuity mistake: When William welcomes the police to the floor, his hand is outstretched. When camera changes his hand is no longer outstretched. (00:06:20)
Episode #1.2 - S1-E2
Continuity mistake: When Richard has concluded that one of the 6 people in the lift killed the bride, the shot changes to show Fidel and Dwayne loading the 4x4 up. Fidel is putting the lights in the back, camera changes and now he's receiving goods off Dwayne. (00:13:30)
Episode #1.2 - S1-E2
Continuity mistake: Camille and Richard are interrogating the victim's parents. The scene begins and the dad has both hands joined, fingers stretched out. Next shot and he has fingers entwined and only a second later he stretches them to assume the correct position. (00:14:15)
Episode #1.2 - S1-E2
Continuity mistake: Richard and Camille are questioning the maid, Margaret. She turns around after answering about the "Cleaning in progress" label, and her braid is down her back. In the rest of the scene, it is in front of her shoulder. (00:18:45)
Episode #1.2 - S1-E2
Continuity mistake: Richard calls it a day on the investigation, but he's still chatting about it with Camille during a walk. When he says "One thing's always puzzled me about weddings", two women are crossing the street in front of him. Few seconds and a cut later, when he mentions the bouquet, here they are again, the same girls crossing the same street coming from the same direction as before. (00:23:20)
Answer: Most likely they had only tropical exotic drinks, he wanted an old fashioned English beer.