Anna cuts the rope, causing her and Kristoff fall down the mountain while they are still tied together. However, in the next scene the rope is nowhere to be found and they are not tied together.
When Anna and Kristoff are thrown down the stairs by Marshmallow, Anna's hat is missing. Her hat is back on her head in the next shot.
Before Anna starts singing "For the First Time in Forever (Reprise)" in Elsa's castle, Olaf is present beside her on the staircase. However as she starts ascending the stairs Olaf is nowhere to be seen.
When Hans finally confronts Elsa, he is not wearing his sword. The sound of his sword being drawn is heard while Anna is on screen, and when Hans reappears he now has his sword in hand but still no scabbard. (Contrast this to the assault on the ice castle where sword and scabbard are clearly visible.)
After Hans' sword shatters he falls to the ground behind a raised patch of ice and neither him nor the block of ice are seen until Elsa raises the ship several minutes later.
Kristoff's stringed instrument has four strings, but only three tuning pegs.
When Elsa is in the prison cell she freezes her shackles and bends them. Frozen metal does not bend, it shatters.
Several times, the characters refer to Kristoff's vehicle as a "sled". Kristoff's vehicle is called a "sleigh" - it is pulled by an animal (in this case, by Sven). A "sled" is what kids use to go scooting down the hill after a big snow.
The wolves have blue eyes. In reality, wolves' eyes can be brown, orange, yellow or even pale green, but are never blue.
During the opening song "Frozen Heart" the ice cutters begin by ramming saws through the ice. Any ice thick enough to support a person, let alone the block thickness seen, would require a hole drilled for the saw to start, the saw would just bend instead.
Gloves were able to hold back Elsa's magic, but it had no problem at all going through her shoes. However, it is possible that the effect of her gloves is merely psychological and that they don't actually stop her powers per se.
When Kristoff falls into the snow after being thrown off by Marshmallow, the snow is perfectly flat on top of him with no dent nor impact marks. However, with deep and loose enough snow, that can happen.
When Anna thaws due to her act of True Love, the group appears to be on a ship even though the characters have not moved. However, when Elsa fell to the ice in reaction to believing Anna dead, there were two nearby masts sticking up through the ice, indicating that a frozen ship was beneath them the whole time.
When Anna first meets Olaf, she soon recalls him from her childhood. But, being as how Olaf was created by Elsa's magic, all memory of him should have been erased by the Troll King. Pabbie said that he would 'leave the fun', so Anna still had memories of playing in the snow with Elsa, but not that it was a result of Elsa's powers.
When Elsa lets her hair down during "Let It Go", her braid passes through her left shoulder. The animators acknowledged this, stating that it did not look satisfactory when they tried to do it physically correct, so they hid this little cheat behind her body instead.
Towards the end of the movie, when Kristoff is holding Anna in his arms, his thumb passes through her torso for a moment.
When Anna and Kristoff first meet Olaf, Anna throws Olaf's head back onto his body. At this point, from our perspective Anna is on the left and Kristoff is on the right with Sven. When Olaf asks why they're hanging off the earth like bats, it shows his upside down point of view. It should show Anna on the right and Kristoff and Sven on the left, a 180 degree Y-axis rotation but it doesn't, it shows Anna on the left and Kristoff and Sven on the right but upside down, an X-axis mirror.
During the song "Love is an Open Door", Anna and Hans are dancing on a light house, and their dancing shadows appear on a nearby sail, but not the shadow of the floor they're dancing on.
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.
During the song, "Fixer Upper," the small troll says, "I don't see no ring." Anna has mittens on, so there is no way you could see a ring.
In the room where Elsa prepares for her coronation and where Hans later leaves Anna for dead is a chess board that is set up incorrectly. The black queen should be on the right of the black king and the white queen should be on the left of the white king from the viewpoint of the players. In the film, it is the other way around. Clearly visible when Hans extinguishes the candle flame.
In the ice palace, when Elsa transforms her garments, her 'sensible shoes' are transformed into contemporary high-heels, which did not exist at the time the film was said to have taken place, circa 1840.
In the film, crossbows and spears are used instead of firearms, even though co-director Jennifer Lee has stated that Frozen's time-line is circa 1840. Crossbows had disappeared from military use by the end of the seventeenth century. Ms. Lee's time-line is thus off by at least 200 years; by 1840, firearms were being used all over Europe.
When Anna is on horseback searching for Elsa, she is seen to be astride the saddle as a man would ride. Young unmarried women of Frozen's time, circa 1840, always rode sidesaddle to maintain proof of their 'purity'.
It is revealed on Elsa's coronation day that Anna had stayed inside the palace for the entirety of the time that the gates were closed, even though the only reason the gates were closed was to prevent word getting out about Elsa's powers. This means Anna could have gone outside and had a normal life whenever she felt like it.
Elsa's ice castle has no food. Idina Menzel's jokey solution to this problem is that she eats 'soy ice cream,' but in the movie she simply has no food.
How did Anna pay for the items she bought from Oaken's shop? As she went after Elsa straight from the ball, it seems unlikely that she would've had any money with her.
Elsa's parents and the trolls knew that the main reason her powers to go out of control were fear. But locking her in her room and keeping her away from people will just add to that.
Why does Anna get applauded when she punches Hans? Whatever his treacherous intentions may ultimately have been, he actually did a good job of ruling Arendelle in Elsa's absence, handing out blankets and soup to everyone. The only characters that know about his true plan at this point are Anna and Olaf (and possibly Elsa and Kristoff). As far as everyone else in the kingdom is concerned, he's the good guy, so their reaction makes no sense.
Several times treason is mentioned as a charge against ambassadors or Elsa. Treason, however, is a crime against the state by a citizen of that state. As such, ambassadors cannot be guilty of treason. Additionally, during Medieval times, the state and the monarch were considered to be one and the same; which means that Elsa, as the monarch, cannot be treasonous.
When Anna is about to go after Elsa, she shouts to bring her horse.
But the horse she rides on is Prince Han's horse and not hers.
Kristoff sings 'reindeers are better than people.' The plural of reindeer is reindeer. He might not be well educated, but as someone who has grown up with a reindeer, and around other ice-gathering teams with reindeer, you would think he'd get at least that right.
In the events leading up to the coronation, a woman explains to her son that "the Queen has come of age". This is, of course, in error; it was Princess Elsa who had come of age, not Queen Elsa. Had it really been Queen Elsa, then there would have been no need for a coronation.
When Anna wakes up on coronation day, and her hair is all a mess, her breasts appear to be Dolly Parton-esque. Throughout the rest of the movie, they appear modest. (Perhaps they assume a corset.)