Plot hole: The Doctor and the Master both surface in bright orange diving suits and are the only two wearing them. But when the supposed body of the Master is dragged out it is also wearing an orange diving suit, and the Doctor and Master are still wearing theirs, so a third suit appeared from nowhere.
Plot hole: In episode 1, on both occasions that Silas P presents his card (first to Daphne S, then later to the Doctor), he gives it to his victims the wrong way around. They have to surreptitiously turn it over so that the side with his name on it is facing them, and then act surprised when he tells them to flip the card over to reveal the words "Happiness Patrol Undercover", as if they hadn't seen it before.
Other mistake: In episode 3, watch the scene where the co-pilot is shooting; you can see his pants rip straight down the seam at the back - which unfortunately happens to be the part of him facing the camera at the time. He then wanders around for most of the rest of the episode with the tear in his pants plainly visible.
Continuity mistake: In episode 2 of "Survival", Midge is captured, and the sabre tooth he's carrying gets knocked to the ground. Neither he nor the Master pick it up before they teleport to Earth. Yet in Episode 3, the Master is able to present the sabre tooth to Midge as he's about to ride his motorbike towards the Doctor.