The Return of Doctor Mysterio - S10-E1
Continuity mistake: When the Doctor pulls a glass of water out of his pocket and hands it to Grant, it jumps from his left hand to his right between shots. (00:05:05)
Continuity mistake: When the Doctor starts writing "Time" on the chalkboard during class, the "T" is slanted, but in a following wideshot when he's completed the vertical "TARDIS" the handwriting has changed. (00:08:10)
Other mistake: After the Doctor tells Bill that Heather's face isn't symmetrical, when he and Bill look into the "puddle" together the Doctor explains that instead of seeing her normal backwards reflection, she's seeing her face "the right way round." The problem occurs in the next shot of the Doctor and Bill seen in the puddle, because the "WOW!" patch on Bill's jacket is properly where it should be, but note everything else is actually backwards. Things such as Bill's color-block top and her denim jacket buttons are on the opposite way. Also note the Doctor's ring has switched hands. Presumably this shot was edited, and the "WOW!" patch was blurred on the left of Bill's jacket then digitally added to the right side, to give the appearance that it's "the right way round" as the Doctor said. (00:20:15)
Continuity mistake: After the Doctor tells Bill that nothing gets through the TARDIS doors, when she responds that they're made of wood and have windows, the Doctor's left hand changes position between shots from being apparently by his side to suddenly resting on the door. (00:28:40)
Continuity mistake: When the Doctor holds out his sonic screwdriver for the Dalek to scan, his grip on it changes between shots. (00:42:25)
Continuity mistake: When Bill says to the Doctor, "Do what you gotta do," he raises his hands to her. His hands are closer to Bill's face in shots facing him than shots facing her. (00:48:40)
Continuity mistake: When the Doctor picks up a skull from the bones waiting to be ground into fertilizer, the position of his hands holding it changes between shots facing him and shots looking at the skull. (00:15:10)
Continuity mistake: After the Doctor and Bill are tied up in the tent of explosives, when Lord Sutcliffe pulls out his pocket watch to check the time and notes that the elephant is being brought onto the ice, in a shot facing the Doctor and Bill the watch chain is taut. When the shot cuts to look at Sutcliffe and his henchman, the chain is slack. (00:33:40)
Knock Knock - S10-E5
Continuity mistake: When Bill walks out of the TARDIS, her teddy bear is lying in the box she's carrying with its head to Bill's right. In the next shot from behind Bill as she sets the box down on the ground, the teddy is suddenly lying the opposite way. (00:06:45)
Continuity mistake: During the Doctor's class he draws lines on the chalkboard, which continually change shape between shots. (00:04:05 - 00:05:10)
Visible crew/equipment: When Bill and Nardole run away at the end of the countdown, the reflections of the camera dolly and camera operator are visible on the CERN countdown board. (00:35:55)
The Pyramid at the End of the World - S10-E8
Continuity mistake: When Erica's eyeglasses get smashed, they're seen in a closeup inside her bag and the lens frame is cracked, but when she takes it out of her bag both lens frames are intact around the broken lenses. Also, when Erica takes the eyeglasses out of her bag she holds the eyeglasses' right temple in the closeup, but in the next shot it's the left temple being held up. (00:06:15)
The Pyramid at the End of the World - S10-E8
Continuity mistake: When the Doctor, Bill, Nardole, et al. first see the pyramid their long shadows are cast directly in front of them, but when the Doctor decides to "bring it" and crosses the barrier, their long shadows are cast directly to their right. Then it keeps changing in the following shots when the Doctor approaches the pyramid. (00:10:45)
The Pyramid at the End of the World - S10-E8
Revealing mistake: The Monks show the Doctor and company simulations of a future, lifeless Earth - but some visible trees reveal the origins of the footage as being from the aftermath of natural disasters. (00:24:10)
The Pyramid at the End of the World - S10-E8
Continuity mistake: While the Doctor, et al. Are in the pyramid with the Monks, when the Monk tells the Secretary General that if his consent isn't pure it will kill him, Xiaolian is standing right behind him in shots facing the Secretary General and shots facing the Monk, but in the one shot of the Secretary General as he's being vaporized Xiaolian has vanished, but Xiaolian is still standing in the same spot in the shot facing the Monk. (00:26:10)
The Pyramid at the End of the World - S10-E8
Visible crew/equipment: Right before the TARDIS materializes in the lab at Agrofuel, there's an overhead shot of Erica at the desk, and the two green tape X marks on the floor behind her are visible, precisely where Douglas had collapsed and his white protective gear should be. Note there's an earlier overhead shot of the same area before the incident, when Erica brings Douglas the coffee, and those tape marks are not there. (00:32:50)
The Pyramid at the End of the World - S10-E8
Factual error: The American officer is called a Colonel but he wears the four star rank patch of a General, and that rank patch is even upside down in some scenes. The stars should have one point upwards.
Continuity mistake: When the Doctor and Nardole are being held by the Picts in a circle of weapons being pointed at them, when Kar's brother points a knife at the Doctor and warns him not to speak until she gets back, his arm is almost fully extended in shots from above and facing the Doctor, but much more bent in shots facing him. (00:12:40)
Continuity mistake: After Nardole gets Pictish facepaint, the lines change in consistency and exact positioning several times throughout the episode.
World Enough and Time - S10-E12
Continuity mistake: After Jorj arrives on the bridge and demands to know which one of the new arrivals is human, the position of Missy's earpiece changes between shots. (00:08:05)
Chosen answer: 1) When Stephen Moffat took over he ignored a lot of what had been developed before (there is not in-universe answer). 2) It would have killed Rose, so the Doctor absorbed the energy. His body regenerated before the energy could do a significant amount of damage that would prevent regeneration.