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.

Death in Paradise (2011)
1 plot hole in Melodies of Murder

Continuity mistake: When Aunt Mary is writing on the whiteboard at the hospital, the word "killed" changes style between shots. (00:31:40)
Episode #1.7 - S1-E7
DI Richard Poole: Everyone had a motive.
Neville : Ahem, you missed someone.
DI Richard Poole: No.
Neville: Me!
DI Richard Poole: Well, you didn't do it...
Neville: Could have!
DI Richard Poole: Really? What was your motive?
Neville: He owed me money?
DI Richard Poole: Did he?
Neville: No.
[DI Richard Poole sighs, shakes his head and moves on]. (00:46:05)
Question: When Richard reminisces about England, he talks about going to the pub and having a pint. Yet the whole time he's in Saint Marie, he is never seen to drink alcohol with the rest of the team; only cups of tea. Why?
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Answer: Most likely they had only tropical exotic drinks, he wanted an old fashioned English beer.