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.
Revealing mistake: Carl, Eddie, and Steve are out ice fishing. At one point Carl falls through the "ice" and you can clearly see that the "ice" is made of some sort of styrofoam. It looks like the stuff used for packaging. The snow looks just as fake.
Continuity mistake: In season 1 Hobbie was 13 and in season 2 he is 10 years old.
The Leap Home (1) - November 25, 1969 - S3-E1
Factual error: The Post Raisin Bran cereal box sitting on the Beckett family breakfast table in 1969 has a 1990s box design. (00:18:00)
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.
Other mistake: In the starting titles, the newspaper clippings state that Doogie sails through high-school in 9 weeks, the article says he is 12 years old. The next clipping says he graduates from Princeton at age 10.
Continuity mistake: Debbie Dobson is looking through a photograph album, the pages she has open have photos of her sister Gemma. The angle changes when Gemma enters the room, just before Debbie turns the page the photos that can now be seen are of Debbie herself. (00:04:45)
Plot hole: When Ellie leaves Ft. Smith on a whiskey boat, she's just been seen with the morning sickness a day or two before. When she leaves the whiskey boat, somewhere between Arkansas and Nebraska, she says she's pregnant but is not showing at all. Even the snaggle-toothed frontier weirdo played by Steve Buscemi remarks that, "You ain't gonna be havin' no baby fer a while yet." The day she gets to Clara's, near Ogallala, Nebraska, she has the baby. Pregnancy lasts nine months. Morning sickness usually begins at about six weeks into it. Ft. Smith, Arkansas, to Ogallala is 561 miles. Even just walking three miles an hour for eight hours a day, one could make it in less than a month. But going via boat and mule-drawn wagon took her over seven months?