When Rose is taking her backpack off to give to Jackie. In the first shot she fully takes it off but in the second shot, it is still swinging off her shoulder.
When Yvonne Hartman is showing the Doctor through Torchwood after leaving the TARDIS, she pushes the door open into the main building with one hand, but when the camera cuts to them entering the building she pushes them open with two hands.
When Yvonne is taking the Doctor and Jackie to the Sphere Room, there is a sign opposite the door that has the words "Torchwood Institute." When Rose is going into the Sphere Chamber, beneath the sign are directions to the Reception, Canteen and Lever Room as well. The whole sign disappears when you see the corridor from inside the Chamber.
At the start when we see Rose on the bus, a bald man sits down on the seat behind her. However, when the camera is inside the bus, he is sitting two seats behind her, without enough time for him to move.
Throughout the series, Rose (and the audience) have had all languages translated for them by the TARDIS. Yet when clicking the television around, all the the foreign correspondents are untranslated.