Character mistake: The whole mystery kicks off when Creek sees a photograph of Alice Spearfish with her parents, and concludes that she is adopted because two people with brown eyes cannot have a child with blue eyes. An obsessive polymath like Creek must surely know that that is wrong. The genetics are complex and though uncommon it is entirely possible for a couple with grey or brown eyes to have a child with blue eyes - and the other way around.
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.
Factual error: During the examination of the crime scene at the beginning of the episode, not one of the Crime Scene Officers or any of the police on site wear latex gloves. The supervising police officer picks up a vital piece of evidence - a small bag of jewellery - with bare hands! They have hopelessly contaminated the scene.
Continuity mistake: When Le Fley runs through the house to be shown that the painting is missing, he is seen in three shots; in the first he is barefoot, in the second he appears to be wearing light-coloured mules, loafers or some other lightweight footwear, and in the third he is barefoot again. It is a little hard to see this in the second shot as his feet are moving quickly, so I could be mistaken, but it does look like he has something on his feet.
Plot hole: The entire plot hinges on Jonathan's solution of the murder of the first wife of Stephen Belkin. That work had the effect of exonerating Belkin. But, Jonathan's solution wouldn't exonerate Belkin in the slightest. It only explained how the murder was accomplished. It did not eliminate Belkin as a suspect. Even after Creek's solution, Belkin still had means, motive and opportunity. As the only surviving person in the locked room where the murder happened, Belkin would have remained the most obvious suspect. In fact, Creek's solution tended to show that only Belkin could have done it, precisely because the room was indeed locked - Creek never explained how a third person could have got in and set up the killing. but in the transmitted episode, it only occurs to Creek much later that, duh, of course Belkin did it.
Deliberate mistake: When Joey and Jonathan get in Joey's car at Metropolis (just before heading to the police station), the car has headrests. When they get to the police station, see Lance Gessler leaving and then follow him, the car no longer has headrests. Obviously removed so as not to obstruct the actors.
Factual error: When it's is revealed that The Omega Man is made of mercury, it is explained that it needs to be frozen to a temp of -38.87c, then it's shown being taken out of the freezer and the mold being removed with bare hands, which would be impossible at that temperature.
Revealing mistake: Creek reads a newspaper article about Tracy, a missing schoolgirl. The first paragraph of the article is about the schoolgirl, but the rest is about John Major (who had just been ousted by Tony Blair as prime minister when the episode was first aired, but, assuming this was filmed some months earlier, would still have been in power). The first line of the article following the first paragraph starts mid-sentence, as if the first paragraph has simply been pasted over the top of the existing article. This mistake is surprising as there is a photograph of John Major alongside the article and the newspaper remains in shot long enough for it to be spotted without the need to pause the DVD or video.