Continuity mistake: When Anna, Kristoff and Olaf get thrown by Marshmallow, Anna does not have her hat on when they are sliding, yet her hat is back on in the next shot.
Continuity mistake: When Anna and Kristoff fall off the cliff after being chased, they are still tied by the rope. But when they land the rope is gone.
Continuity mistake: In "Let it Go", after Elsa releases her cloak, an overhead shot shows she's about 30 feet short of a gap, walking forwards, on a flat expanse of snow. She then keeps moving towards it while singing in closeup (which would have taken her right up to the edge). She then runs forwards in a wide shot, but instead of falling off the edge she should be right next to, she runs up over a hill that appears from nowhere, before she creates the steps to bridge the gap.
Other mistake: Anna is shown from the front as she ascends an ice staircase inside Elsa's ice castle. There are giant ice pillars to Anna's right and left. They should reflect the side of her face, but instead the reflections show her face from the front.
Continuity mistake: Anna walks a few feet to put Olaf's head back the right way on his body. After she does this, she is back to the position she started in and Olaf has moved between Anna and Kristoff.
Revealing mistake: When Anna and Hans are dancing in the spotlight, their silhouettes should be casting shadows as well, but aren't.
Continuity mistake: In the scene where Kristoff and Anna are riding on the sled, in the first shot of them in the sled, Anna's hair is behind her, flowing in the wind. In the next shot, her hair is in front of her, resting on her chest.
Continuity mistake: During the song "For The First Time in Forever", Anna runs into the gallery and a painting of a woman on a swing is seen to the right of the door. Later in the scene she is seen leaping off a teal-colored seat to make the same pose as the woman. This is impossible since we never see her cross in front of the door to get to this painting, which means she would have had to have jumped very far to get to the swing painting.
Continuity mistake: When Anna is singing "For The First Time In Forever", at one point she is walking along the railing of a bridge. As she is doing so, she passes a guy in a hat, holding his daughter. In the next shot, the guy and daughter completely disappear.
Continuity mistake: In the shot where Anna and Kristoff gets a first glimpse of Elsa's castle, the camera starts from behind an exposed rock protrusion, but in the shot of the mountain before the song "Let it Go" starts there was none.
Plot hole: Hans' plan makes no sense because that's not how being royalty works. If he married Anna, then Elsa was dead, Anna ascended the throne, then Anna was dead, he would not remain king. The crown would be passed to a relative of their family or to another high-ranking noble family within the kingdom. Hans would always be a prince no matter what.
Continuity mistake: Just before the end Olaf gets his own personal fluff (a snow cloud). In the scene on the ice in the courtyard following that, the snow cloud is not there.
Continuity mistake: When Anna enters the trading post there is a target next to the door that is facing the same direction as her, but moments later the target is facing Oaken's counter without being touched.
Continuity mistake: The snow covering the mountain varies between shots of the first sighting of the ice castle and when Sven attempts to climb the stairs.
Continuity mistake: When Anna blocks Hans from hitting Elsa with his sword, Hans flies backwards. If you look at the spot where Hans should be, he is not there.
Continuity mistake: Towards the end of the scene where Hans is about to kill Elsa, the camera pans to a wide view of Anna standing on the ice on her own, Elsa and Hans are nowhere in view, yet a few seconds later as Hans brings his sword down on Elsa, Anna appears and parries the blow. In her fragile and barely able to walk state, there is no way Anna could have covered this distance in the time given.