Plot hole: In Episode 1-7: The Serene Squall, the Enterprise is at the edge of Federation space, and it is stated that it would take two days for a message to be received by Starfleet. Later in the episode, they have no problem establishing a real-time connection to Vulcan, which, according to canon, is only 16.5 light years away from Earth.
Plot hole: The powerful 'amulet of protection' makes the bullets strike the shooters instead of John. Clearly it works not just against magic and other ethereal threats, but also against physical attacks. So why does he almost get knocked over by a car? True, he isn't seriously injured, but he was thrown to the ground. The amulet should have pulverised that car or bounced it at least two blocks away.
A Deadly Net - S2-E2
Plot hole: S2E2 A Deadly Net Midway through the episode, Sylvia and others in the cast bid people goodnight, offer after-dinner drinks, but the sun is shining brightly and continues to do so for quite a period of time as events continue to unfold.
Woman in the White Hat - S2-E4
Plot hole: Ronnie and her daughter, and son-in-law spot a pick-up truck towing a large trailer through her plantation. Ronnie runs to her truck to investigate. When she arrives, which could only be a few minutes behind the truck and trailer, there are technicians well into the process of mixing chemicals.
The Mystery of the Yellow Room - S1-E8
Plot hole: The resolution of the case hinges on the fact that the titular room had a door that was then sealed off completely by order of a certain character. The key word here is "order"; the culprit did not do it by their own hand, they ordered other people (normal White House staff being asked to perform normal work) to get it done, without even overseeing the operation. They did it to conceal an incriminating object that was inside the closet that was being sealed. No worker (or team of workers, working in a historical building such as the White House, even) in their right mind would wall up a closet without first opening it and making sure it was empty. From the culprit's perspective, they directed a crew of people towards the place where the very conspicuous, incriminating, bloodied evidence was hidden and just assumed nobody would carry out due diligence. And it worked.
Plot hole: Ep 6 Ffion, with backing by a squad of police, is in a frantic search to find the killer's latest victim. She finds something and calls into headquarters, whereby the DCI says, "I'll send backup." She already has a swarm of police aiding in the search.