Before the opening credits, Clara puts her head through a hatch in the roof of The Doctor's carriage. In the first scene after the credits, it can be seen that there is no hatch in the roof.
The spiral staircase changes direction between the long shots and close ups. When she starts to walk up it the out edge is on her left but is on her right when she reaches the top.
When The Doctor helps Clara out the window on the roof and they fall over, his hand goes from being in the center of her back to on her waist between shots.
When The Doctor opens the door to meet Dr Simeon, there is a light above him. But the light is gone in the next shot, when The Doctor closes the door.
When The Doctor and Clara are running away from the ice-governess and climb the stairs to his Tardis, they start climbing the stairs clock-wise, then climb it anti-clockwise when they get near the top, even thought the ice governess is still climbing them clockwise. The stairs literally flip the opposite way.
When Clara lands after pulling the ladder down in the park, the ground moves as if the snow was sitting on top of a plastic sheet.
When The Doctor opens the door to meet Simeon, there is light above him. In the next shot, once he has closed the door, the light is gone.
The children are shown with teddy-bears, though these were not invented until ten years after the story is set.