In "Let It Go," Elsa refers to her ice magic as "frozen fractals." Although the concept of fractals has been known since the 17th century, the actual word "fractal" wasn't coined until 1975, and thus nobody in Frozen's time period would have known it.
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.
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.
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 Sven and Kristoff are leaving Arendelle after delivering Anna, there is a shot of Sven looking back where Arendelle should be, but it is missing from the shot. When Kristoff begins running down the hill Arendelle appears again.
Outside Wandering Oaken's Trading Post there is no evidence of wind or storms. However, when the door opens for both Anna and Kristoff's entry, the sound of blowing wind can be heard and the wind apparently slams the door.
When Kristoff leaves Arendelle after leaving sick Anna on the castle, Elsa escapes and releases a blizzard on Arendelle, making Kristoff come back, but he takes way too much time to get back to Arendelle compared to the amount of time he took to leave.
When Prince Hans is encountering Marshmallow for the first time, the two Gendarmes are tossed against a snow bank after they both fire their crossbows. As they are recovering, they collect themselves and you can see the crossbows on the ground, but no bolts, as one of them spots the Queen running up the stairs. In the next shot a bolt appears under the Gendarmes as he picks it up and chases after the queen.
When Anna enters Elsa's ice castle, she slips on the ice and catches her balance. At this point she is standing well into the middle of the room. When the view cuts to a front view a moment later, she is standing at the base of the staircase.
When Kristoff & Sven have left Anna at the castle and go back to the mountain, they're stepping in the snow (leaving footprints). When the cloud appears over the castle and they return, there are no signs of their footprints.
When Anna and Kristoff are in the sled on the way to the North Mountain, Kristoff asks her, "Didn't your parents ever warn you about strangers?" Anna replies, "Yes, they did," and moves over. The camera then focuses on Kristoff and then both of them. Anna is then shown in her original position.
As Elsa runs out of the castle after the coronation party, the fountains freeze and then the fjord freezes as she runs across it. After Anna's horse gets scared and runs off, Anna falls down a hill and into a stream where her clothes freeze. When Kristoff and Anna enter "wonderland" right before they meet Olaf, there is a scene where there is a frozen waterfall. If everything else in the kingdom is frozen, there shouldn't be free flowing water where Anna would fall into and freeze.
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.
After Olaf's song ("In Summer") Hans hands a bunch of cloaks and warm clothes to a guard back in Arendelle to be given away. After he threatens the Duke of Weselton, Anna's horse returns to the castle and startles the same guard who drops the clothes. The camera then moves to a medium shot of Hans calming down the horse, and then returns to a full shot of Hans, the Duke and some villagers and the clothes the guard dropped disappear for the remainder of the scene.
When Anna first meets Kristoff at the trading post, he mentions his ice business and it shows his sled full of ice blocks. The next time you see the sled is when Kristoff and Anna are riding to the north mountain with no ice blocks.
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.
When Anna and company are trudging up the mountain before meeting Olaf they are walking from screen right to screen left. Anna stops and turns to her right, away from the camera, to look at Arendelle. She and Kristoff have a short conversation, then she points to her right, back the way they just came, and asks "This way to the north mountain?" Kristoff makes a smart comment and the camera pans to Anna's right and shows the north mountain.
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.
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.
After Olaf jumped off the cliff, Princess Anna and Kristoff fell after cutting the rope. Olaf began grabbing Kristoff's feet, fearing they were his own. Kristoff then sat up from under the snow, showing Olaf was sitting on Kristoff's legs. This wouldn't have been possible if Olaf fell before Kristoff.
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.
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 gate 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.
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.
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.
When Elsa lets her hair down during "Let It Go", her braid passes through her left shoulder. The animators acknowledged this, stating that it it did not look satisfactory when they tried to do it physically correct, so they hid this little cheat behind her body instead.
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.
The goof items below may give away important plot points.
When Prince 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 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.)
In "For The First Time In Forever" Elsa's study is located at the end of a long hallway. As Olaf and Anna leave the study after Hans betrayal, it has moved to the middle of a long hallway, with the doors located across from a window.
In the prologue, the royal parents quickly bring the girls to the trolls. But later when Kristoff brings Anna back to Arendelle from the trolls, he takes longer than the guards did to capture Elsa and take her back to Arendelle from the North Mountain, which is further from Arendelle.
When Anna becomes fully frozen, the clothes she wears become frozen too. Everything within a very close range froze also. Prince Hans' sword was close enough and began to freeze as well, which is why it shattered when it hit Anna's hand.
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 through 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.
Hans's plan is to marry Anna and then take the throne by killing Elsa and Anna. A spouse of royalty cannot take a throne on their own, instead it passes to the next person in the order of succession as it was before the marriage. So the plan is flawed from the start, and Hans would know this.