During the first verse of the song "I have A Dream" (in the "Snugly Duckling"), the hook-hand thug hooks the keys out of the piano and they are sent flying across the room. In the next shot, he is continues to play the piano as if the keys were still there (which of course they can't be).
In the first scene we see Mother Gothel, she is approaching the Magic Golden Flower wearing a blue dress and a red cape. However, when we come back to this same scene just a few seconds later, she is shown wearing a red dress and blue cape. The red dress and blue cape coloring remains throughout the rest of the movie.
All ships in the harbor are clearly at anchor and unmoved despite the light wind (clearly seen in the lanterns scene). Yet, all of them are in full sail, which is clearly a mistake, even for an animated fantasy movie.
When Flynn and Rapunzel are in the cave, it is filling with water, despite the fact that their heads are in a small pocket of air higher than the water level. There's no way the water could fill up to cover their heads if there was nowhere for the air already in the cave to go.
When Mother Gothel goes into Rapunzel's tower via the secret passage after meeting Maximus, Rapunzel's room is dark because the window and its curtains, and the roof, are closed. But when Rapunzel left her tower for the first time, the roof and the window were open; it was light inside. The window, curtain and roof couldn't have been closed after Rapunzel left, there was nobody else in the tower.
Rapunzel wants to leave the tower to see the lanterns that appear only on her birthday. If Gothel had given Rapunzel a different birthday she might not have been so determined to leave. Unfortunately for mother Gothel, she didn't know about the lanterns until Rapunzel told her just before turning 18, and thus couldn't imagine that retaining the birthday would have any significance.
Rapunzel's hair is just long enough to pull Mother Gothel up, but it is also long enough to lower Rapunzel to the ground, which would take twice as much hair. However, it is seen that Rapunzel loops her hair over hooks when pulling Gothel up, which would allow for half the hair to still be in the tower, with just enough lowered to serve Gothel.
When Rapunzel and Flynn arrive at the lantern festival and cross the bridge, Rapunzel looks back to the camera and exclaims "Wow!" but there is no voice to the word. Actually, she's not exclaiming - she's mouthing the word silently, because she's so stunned by the view.
Rapunzel's "Mother" sees the light reflecting off the crown, and she knows that Rapunzel is gone. At the end of the movie, Rapunzel gives Flynn the crown. If she took it with her, then the mother would not have seen it in the tower. Mother Gothel gives the crown to Rapunzel to give to Flynn.
When Flynn is hiding from Maximus in a cave, Maximus can be seen looking behind him. Horses have to turn their whole neck to look behind them, but Max turns his head like an owl would. However, Maximus exhibits "super-horse" and non-horse abilities and behavior throughout the movie, most often those of a dog. Such discrepancies are just part of his character, to make him funnier.
The goof items below may give away important plot points.
When Flynn is dying, he is on his back with his right hand palm up on the floor. Later when we see both lovers from far above (when the magic is working) his right hand is palm down. If Flynn is dead, he wouldn't be able to change the position of his hand.
As "Mother" stabs Eugene after he climbs the tower to rescue Rapunzel, the knife comes out clean, no blood. However, blood on the knife would likely be reason to increase the MPAA rating to a PG-13, so to keep the rating at a PG, the blood was omitted.
When Eugene cuts Rapunzel's hair at the end of the film, the hair still connected to her head also turns brunette like the hair that was cut. However, it's established in the film that the hair connected to her roots stays blonde, and only the parts cut off from the roots turn brunette. Therefore, her hair on her head should have stayed blonde despite most of it being cut. However, when Gothel cut a lock of hair early on we see the hair turn brown where it was cut as well as the lock itself. Also, Rapunzel shows this part of her hair to Flynn while they're sitting by the campfire.