The Mysterious Affair at Styles - S3-E1
Plot hole: Can't fault this massive plot hole to the adaptation, but to the source material; the culprit (forgetting the stupidity of writing an incriminating letter detailing the plan to murder someone, and put it in a desk he shares with her) since there are people outside the room that are about to enter, tears the letter in 3 neat vertical strips, rolls them, puts them in the vase on the mantlepiece, and then opens the side door to slip away...instead of simply pocketing the letter and going through that same door. Nobody was going to search him or anything and could have burned it, torn it into confetti, anything, later. It takes way way longer to do what he did, which needed him to stay there in the room increasing the chances of being found out. And of course he and his accomplice do not retrieve the letter after.
Plot hole: The Winslows' younger daughter, Judy, disappeared from the show in 1993 (halfway through the series' run) with no explanation.
Another Mother - September 30, 1981 - S2-E13
Plot hole: Sam wallops both kidnappers and rescues the son. They then leave the two child molesters unconscious beside their van. No one calls the police or has them arrested, so they're free to kidnap another kid. Sam doesn't leap until after an epilogue in which the loose ends are supposed to be resolved - but this question is left unanswered. It's not excused by a "just because we didn't see it happen doesn't mean it didn't" defense. The viewers shouldn't be left with a dangling plot hole, and one line could have cured this one.
Plot hole: If the electrical lights go out at a quarter past ten at Guardbridge, how come there is bright electrical (as suggested by it not flickering like candle light) light coming from a room behind Hal Stanton and Catherine Faulkner when they kiss? This scene takes place after 22:15.