The Sinner and the Sandman - S5-E2
Plot hole: Ian Avery-Cooper loses his lottery ticket which is immediately picked up and appropriated by Leonard Corbyn. However, Avery-Cooper immediately reports the loss to the shopkeeper who sold him the ticket. All lottery sales staff are trained in what to do under these circumstances, because it happens a lot - they cancel the lost ticket and issue a new one. Since Avery-Cooper used the same numbers every week this would not pose a problem, but even if he didn't the ticket would be recorded on the seller's computers and could be precisely reconstructed. This would be even easier since Avery-Cooper reports the loss within a minute of it happening and we see that he has the receipt for the sale in his hand. Corbyn's stolen ticket would be worthless and Avery-Cooper would be able to claim his rightful winnings. If nothing else Ian could have simply bought another ticket with the same numbers.
Plot hole: Throughout the show it is reported several times that Billy and Ally met when they were both eight years old in a park by sniffing each other's bottoms. However, when Billy is buried in season 3 his tombstone reads, 1968-2000, making him 32 years old. Ally is only 29 and does not turn 30 until later in that season which would at the very least, put them 2 years apart. (00:41:15)
Plot hole: In the Pudding Club episode, when they visit the secret room, all the candles are lit even though no-one is in there.
Hiren Lake Legend Murder Case: File 3 - S1-E6
Plot hole: The plan of the killer relies on a couple of pretty strong coincidences, such as the fact that they wouldn't realise their common link (he knows nothing about what happened on the cruise, so he couldn't know that they all wouldn't realise they have met before) and the fact that the boat is discovered at the perfect time for him to commit murder (on someone who wasn't part of the trip, but is willing to join him).
Plot hole: When Mike Walker is out looking for his son Connor calls Satch and asks, "What's up with Mike?". She couldn't know that there was anything up with him, she left before he heard his son was missing. Had she called Walker on his cell, he would have explained what was going on, as he was very eager to tell her. Obviously a set-up so that Satch could tell her that the boy was missing.
Plot hole: During Ryder's report, as Bruce and company watch it on TV, the camera suddenly zooms in on Joker standing on a catwalk above Ryder, and none of the crew, especially not the camera man who caught the villain, makes a comment about this. They may have thought the Joker (or rather an impersonator) was perhaps a surprise gag in the show, but since this is supposed to be a serious documentary report, it is still strange that they wouldn't point it out.