(at around 57 mins) The British commander demands a witch to interpret Carina's calculations on the prison cell wall. In reality, a naval commander of the 18th century - or any educated person at the time, in fact - would be quite familiar with such drawings. Carina's abilities shouldn't have been surprising to anyone, much less a sailor, as astronomy was a proven science and the British in particular very keen on its study. In fact, by the time the movie is set, the Greenwich Naval Observatory was over a century old. At the most, Carina would have been considered eccentric or a little odd, but never a witch.
(at around 1h 28 mins) Carina says she is named after "the brightest star of the North". This is wrong for several reasons. First, Carina is not a star, but a constellation. Secondly, it is a southern constellation, in fact it can't even be seen in most of the Northern hemisphere. Finally, the movie takes place in 1755, and Carina wasn't named until 1763, having been an integral part of the larger Argo Navis constellation.
Carina Smyth's trial and conviction for witchcraft is not realistic of the time period. The last execution for witchcraft in England was in 1716, with the Witchcraft Act of 1738 completely abolishing the practice in 1736 all over the British Empire. The movie is set in 1755.
(at around 43 mins) As Jack is mentioned the guillotine as a form of execution, he says it is French and mentions that they also invented mayonnaise. While he is right, the guillotine was given its name and current appearance (particularly the angled blade shown in the movie) by Dr. Joseph Guillotin in 1789. While a previous version was already available in England at the time the movie is set (1750's), it was called a "maiden", and had a curved or straight blade. Also, mayonnaise sauce, though previous versions were known, was not given that name until 1806.
(at around 19 mins) When Carina Smyth uses the telescope and the steward says: "No woman has ever handled my herschel". Wilhelm Herschel would have been 17 years of age in 1755. Wilhelm Herschel is famous for a new type of reflector, called herschel telescope, but visible telescope is a refractor.
(at around 1h 9 mins) There is no way that Carina Smyth would carry a pocket chronometer. Maritime chronometers were invented around 1761 though earlier attempts were made. But no chronometer could be carried in a hand. They were large instruments in a gimbal mount in a wooden box.
(at around 1h 27 mins) An order is given on a British naval ship to turn to port. The correct term for the period is larboard. The Royal Navy did not replace the term "larboard" with "port" until 1844.
Gunlocks, or flintlock firing mechanisms, are visible on the cannons of the British ship as an officer cocks them. These devices were rare since they only could be adapted to new guns, and only became a standard issue in the British Navy around the 1800's, fifty years after the movie is set.
(at around 1h 3 mins) This film shows the compass being given to Jack by a captain he worked for in his youth as that captain is dying. In Pirates of the Caribbean: Dead Man's Chest, Tia Dalma said he bartered the compass from her.
After Jack falls into the pig pen he, and his compass are covered in mud. When he visits the pub and offers to trade the now filthy compass for a bottle of alcohol, it suddenly appears spotlessly clean when tossed into the barman's basket.
(at around 14 mins) The first scenes take place on the island of Saint Martin that appears to be under English rule. This is historically inaccurate for Saint Martin has been under Dutch and French rule since the mid 17th century. This movie is set in the mid 18th century.
When the crew leave Jack after the bank job, they walk along the beach in front of the docked ship where there is an island visible in the background. When they release the ship into the sea later, there is nothing but open blue ocean expanse surrounding them and no island.
(at around 35 mins) Jack still refers to Will as a eunuch. Of course Will is not a eunuch, as he managed to father a child. Jack referring to Will as eunuch is a recurring joke from the first movie, not to be taken seriously.
It is said that Salazar leaves one man alive to tell the tale, but during the first battle he kills the captain of the ship and only finds out there is Turner below the deck when he trips over a bucket. Which means Salazar didn't know anyone else were below deck yet he killed the Captain which according to Salazar should've been the last man.
The goof items below may give away important plot points.
At the climax, the characters escape by climbing the anchor's chain as it is being pulled up. Yet, during the age of sail, thick hemp cables covered in tar were used. Chains were too heavy and would rust easily and therefore snap because iron manufacturing was still too primitive, and were not used for hoisting anchors until after the Second Industrial Revolution (after 1850), when iron had been perfected and steam machinery was in use.
The destruction of the Trident at the end of the film broke all the curses of the sea, which meant that Will Turner is no longer bound to the Flying Dutchman. However, at the end of Pirates of the Caribbean: At World's End, his heart was cut out and put into the Dead Man's Chest. Therefore, when the curse broke, he should have dropped dead. The same applies to Salazar and his crew, as it is shown they all died during the explosion of their ship. Evidently, the Trident doesn't merely break the curse, but also restores the cursed to their original form. This can also explain the appearance of Davy Jones, whose heart was not only cut out, but stabbed.
(at around 35 mins) Some may think that Miss Elizabeth Swann should be introduced as Mrs. William Turner. She and Will were married by Captain Barbossa at the height of the sea battle with Davy Jones. In fact, Barbossa greets her as Mrs. Turner. However, it is a commonly mistaken belief that ships' captains may legally marry couples. They cannot, legally. It is Barbossa's joke to call her "Mrs Turner."
(at around 1h 30 mins) As Carina's origin is discovered, Jack taunts Barbossa by saying "Smyth is not a common name". Actually, Smyth is just an alternate spelling of Smith, the most popular surname in the English language. But Jack is obviously being sarcastic.
(at around 2h 5 mins) In the post-credits scene, when Davy Jones' claw rises up, it seems to be on his right arm due to Jones not being visible behind it. However, the claw has been on Jones' left arm; if that were the case in this scene, Jones would be visible behind his arm. But there is some ambiguity as to whether this is really Jones or just a dream of Will Turner.