When Jack first begins to run inside of the giant wheel, he bumps his head on a crossbeam. When they are all running inside the wheel later, you can see that the crossbeams are "higher" to the middle of the wheel. He shouldn't have been able to bump his head.
When Jack Sparrow, Elizabeth and Norrington are arguing on the pier in Tortuga, the hull of a ship can be clearly seen behind them bearing a Plimsoll line, which wasn't used until the late 19th century.
When Elizabeth kisses Jack Sparrow and handcuffs him to the ship at the end of the film, we hear the distinct ratchet sound of modern handcuffs being closed. In the following close up shot of the cuffs, we see that they are consistent with irons of that era and could only have been put on with a hammer and anvil.
Gibbs hints to Will that the crew's bone cages were made from the bones of the Pearl's missing crew members. However, those same bones are far too large to be human. The bones are also too aged to have been made from the rest of the crew.
At the beginning of the movie when Master Gibbs is telling the tale of the Kraken, Masters Pintel and Ragetti appear to have no knowledge of the Kraken, but later in the movie, when they are rowing the boat ashore, they have a heated discussion as to the proper pronunciation of the Kraken, as if they had known it all along.
As Jack is rowing away from the Pearl, he turns around to look at the land behind him. As he is doing so, he is holding his compass and it's open. When he turns around to face the Pearl again, he reaches down and pulls out his compass.
When the Flying Dutchman first gives a broadside to the Black Pearl, it does a fair amount of damage to the Captain's cabin stern windows. Yet when the Pearl is fired upon by the Dutchman using the "triple guns", the damage isn't there anymore.
When Jack escapes the island from the Cannibals and is hit by a wave, you can still see the eyes vaguely on his face, however, the eyes that were painted over his eyelids that made it look like his eyes were open when they were shut are completely gone with no trace.
When Will is on the half-sunken ship and fighting Jones' crew, he slashes a crew member and fish gush from his body. Almost five seconds later, in the same spot he is hit on the head and falls to the ground. The spilled fish are nowhere in sight.
When Jack is running from the natives, in the first close up shot you see a rope tied around Jack waist. After the next shot of him running to the ship you do not see it any more and he had no time to take it off.
On the island, when Elizabeth has forgotten her sword, an axe is thrown towards them and hacks into a tree. The axe comes from a totally different angle, than where the monsters are coming from. The monsters are coming from the right, the axe is thrown from the left.
When Jack is about to be roasted, the native who runs across the bridge with the fire has sticks going through his right ear. Then they do a close-up the sticks are in his left ear. Then in the next shot they are in his right again.
Near the beginning of the film, when Will is brought in in shackles, Gov. Swann appears with nothing in his hands and demands the guards to stand down. Cutler Beckett hands him the first warrant - the one allegedly for Will - and suddenly Gov. Swann already has a warrant in his hand. As the scene continues, he has anywhere between three and four pieces of paper in his hand for the two warrants. (This error may be related to the fact that in the script Beckett had several lines of dialogue cut from the final film, explaining his appointment to Port Royal, for which the extraneous papers may be related.)
The first time that Will is fighting the Kracken, he jumps to a sail, stabs his sword in it, and slowly goes down the sail. In one shot the material that makes up the sail is sewn together sideways, as is factually correct for how sails were made at the time, but when he goes down the sail the material is put together up and down so that his sword does not snag on the seams.
When Davey Jones first comes on the Black Pearl, Jack Sparrow has a sparkly coin/disc thing in his hair on the right (your left). As Davey Jones and Jack are talking and the scene switches back and forth the coin/disc Disappears then reappears again in his hair.
When Will is about to be whipped, two of the Flying Dutchman's crew tear his shirt while leaving the collar intact. You can see the collar against Will's hair as the two haves are wrapped over Wills arms to expose his back. As the first blow lands the shirt is hanging completely lose with the collar torn through.
When the bone cage containing Will, Gibbs, Cotton, Marty, etc. falls into the water you see the back of Cotton throwing it off of himself. In the very next shot you see Cotton crawling out of the cage after everyone else. He would have already started swimming and not still be in the cage.
Right before Jack removes his handcuffs, his rings are pointing towards his palm to prevent them from getting in the way, but in the scene right after that, when he is free, his rings are pointing away from his palm, the correct way to wear rings.
There's a common misconception that references to Singapore in the series are anachronistic, based on the erroneous notion that the island was named by Sir Stamford Raffles in 1819. It was actually named "Singapura" by a Malay prince in the 14th century.
After Davy Jones wipes the "black spot" from Jack's hand, Jack looks at it and you can see a black mark still visible on the back of his hand between his thumb and finger. The dirt on his hand between his thumb and index finger is there to cover up a tattoo of Johnny Depp's. It is not part of the black spot, which is only on the palm.
When Will and Jack are looking out on the Flying Dutchman (Will holding a lantern) two people walking behind them are cut in half by the green screen background. In other words, their heads float by on air.
When the large water wheel with Norrington and Will Turner rolls into the water where Jack and Elizabeth are fighting Davey Jones' crew, it does not leave a wake when it rolls past Elizabeth and falls over on its side. It must be CGI, as an actual body moving through water would have left a wake.
When Elizabeth tries to head for the gun in order to shoot the gun powder (and rum) barrels she tries to climb some stairs (right before Jack takes up the gun). Suddenly, the ship is hit so hard that Elizabeth falls down. The gun, however, does not move at all.
The goof items below may give away important plot points.
When Tia Dalma is asking Will and Elizabeth what they would do to bring back Jack Sparrow, the mug on the plate in her hands moves from between the candle and herself, to a position on the left of the candle (from her perspective) between shots.
The curse of the Aztec Gold turns those who stole from it into undead beings. William "Bootstrap Bill" Turner, Sr. tells Jack that he was offered a way out of the crushing depths, were he would have spent an eternity, by Davy Jones. However, in keeping with continuity from the period and revealed in DVD extras, most sailors of this period were unable to swim. Being unable to die, serving aboard the Dutchman could be viewed as an escape.