Continuity mistake: When Rory finds his dad on the spaceship, Brian is lying on a gurney next to the two farthest slabs, which are empty. When Amy and the Doctor come through the portal to the spaceship, Brian's gurney is between the two closest slabs, with Rory lying on the closest one, which previously had a completely different person lying on it. (00:35:05)
Other mistake: Billie Piper is listed as playing "Rose" in the closing credits. Rose was certainly mentioned, yes, but Piper in this episode was actually sentient superweapon the Moment, who merely took Rose's form, and was in all other respects nothing like the former companion.
Continuity mistake: Among the various contents of the Black Archive are visible several objects that shouldn't be there, because it is impossible for U.N.I.T. to have acquired them. In particular: A detonation pack from "Planet of the Ood", set in the 42nd century, a few thousand years in the future. The Supreme Dalek's shell from "The Stolen Earth"/"Journey's End" - it was shot and killed by Jack Harkness, destroying the shell. The restraint chair from "The End of Time", last seen on the Vinvocci's ship. The makeshift sonic screwdriver built by the alternate Amy from "The Girl Who Waited", since that version of Amy was wiped from existence. A different explosive carried by Gregor Van Baalen in "Journey to the Centre of the TARDIS", which is both set in the future and involves a complicated situation in which the events ended up technically not happening.
Other mistake: As the Doctor passes a stack of three cubes, the camera is close enough to see the top cube has two small squares intertwined on the side facing diagonally toward the light. He said all were the same, no difference or individuality whatsoever. (00:03:53)
Continuity mistake: After the 10th Doctor marries Elizabeth, they kiss. While they are kissing, the collar on his jacket gets turned up, from her point of view. The camera then switches to look at her from over his shoulder, then back, and now the collar is straight. (00:49:40)
Continuity mistake: At the end, when the Tenth Doctor steps into his TARDIS, there are three sets of cups and saucers on the side of the bench farthest from the TARDIS, and Clara's set in the middle of the bench, but as the TARDIS fades away, there are two sets on each end of the bench. (01:14:10)
Continuity mistake: When the Doctor is at the Tower of London with Amy and Kate, he claims he needs some air. When Amy follows him she follows him twice, in two different shots. The first time on 'ground view' walking and the second time on 'above view' running. (00:21:20)
Plot hole: At the beginning, when the Doctor, Clara, Angie and Artie get off the TARDIS, the two kids are absolutely convinced that they're on the Moon, despite the Doctor explaining that they're actually at the Spacey Zoomer ride on Hedgewick's World, the greatest amusement park in the universe. The kids only realise they really aren't on the Moon when Webley emerges from inside of a fake rock formation, followed by the soldiers. The part of the room that the soldiers come from, which should have been fully visible to the time travellers from the beginning - in fact, they should have been looking directly at it when they leaned out of the TARDIS in the very first shot - looks nothing like the Moon and contains a sign for the Spacey Zoomer ride and a large map of the amusement park. (00:00:01 - 00:00:45)