The movie has Barnum start his circus at a rather young age. Actually he did not start his famous circus until he was 60. Also the circus did not start until five years after his museum burned down in 1865.
When the front of Buckingham Palace is shown, the Union Jack is flying above the castle. Until 1997, the only flag to ever fly above Buckingham Palace was the Royal Standard, which is only flown when the sovereign is in residence. Following the death of Princess Diana, Queen Elizabeth II was under great pressure to break with tradition and fly the Union Jack at half mast, which she did. Since then, the Union Jack only flies above the palace when the Queen is not in residence or at half mast upon the death of a member of the Royal Family.
When Barnum is still working at the shipping company (1841, before the museum purchase), he mentions the German, Lilienthal, and his gliding experiments. Lilienthal wasn't born until 1848 and didn't start flying until 1891.
When the thug throws the lantern, it hits a poster that promotes a wax figure of "President Lincoln". Barnum was returning from his tour with Jenny Lind when this scene happened. The tour took place in 1850, ten years before Lincoln was elected President of the U.S.
Early in the film, P.T. and Charity have just arrived in New York City, and we see an image of the Flatiron Building, under construction. Construction of the Flatiron Building began in 1901, ten years after Barnum's death. It was completed a year later, in 1902.
Barnum's marriage is endangered when a press photographer snaps the instant that he & Jenny Lind kiss; but this was before the development of fast film, when people had to pose for a while because anything which moved became a ghostly shadow. [Update: while pictures were taken of the event, the newspaper stories show a drawing instead of a photograph which is correct for the time period. It was the story that endangered the marriage and not a photograph.]
Barnum and Charity dance on the rooftops early in the film, depicting the start of their lives together in New York. This would haven been in the 1830s, yet the Brooklyn Bridge is seen under construction in the background. Construction on the Brooklyn Bridge did not start until 1869.
While the film is to be commended for using what appears to be a real orchestra in the Jenny Lind concert scenes, the double bass players have low C extensions which weren't invented until the turn of the 20th century and didn't gain popularity until the 1920's.
The front elevation of Buckingham Palace shows the 1913 remodeled exterior, whereas during the period of Barnum's visit, a different facade was seen on the Palace, which did not include the now famous balcony.
When Barnum apparently opens his museum to the public, patrons purchasing tickets using Indian Head (nic) cents and nickel three cent coins; both were released 1859,1866 respectively. The scene in the movie is before the Jenny Lind scene (1850) Therefore the coins should be large cents and silver half-dime coins.
Barnum and Carlyle each down 7 shots over 4 minutes during "the Other Side" and can still dance their way back to the circus and up a set of stairs to the trapeze level. Even if they are both experienced drinkers they should be showing some physical impairment.
The actor who plays Tom Thumb is 4' 2" in real life so he's made to look shorter in the film by having the actor stand or walk on his knees with fake shoes attached. In the 'This Is Me' number, Tom Thumb appears and disappears at different points. He's in the group of oddities as they enter the hall of concert goers since he reaches up for the Bearded Lady's hand and he sings his line. He's missing from the overhead group shot as the oddities march out of the concert hall towards the circus, since the actor would have been unable to keep up with the group shuffling along on his knees. He's still missing as the group approaches the monument outside the circus. Then he suddenly reappears when the group is stationary singing at the monument.
At the last bar scene when all the circus crew are dancing, you can see in the back how P.T grabs his hat and starts running for the door to reunite with his wife but then it cuts back to the same scene close up.
When the young P. T. Barnum and his father are leaving Charity's home he is carrying fabric which include yellow fabric bolt and a purple patterned fabric bolt, when he spills them down the stairs those fabrics are different colors.
When Anne is in slow motion on the trapeze the first time she and Phillip see each other, her left arm is out to her side. The arm remains out to her side as their eyes meet, but in the next clip just before it speeds up to normal, both of her arms are out in front of her.
During "Rewrite the Stars" Zendaya is using one rope to do all of her stunts on. When her and Zac Efron come together another rope appears out of nowhere that they both swing on. When the song ends there is only one rope.
During the last song 'The Greatest Showman' when P.T. Barnum goes off and swaps with Carlyle, as Carlyle runs towards the ring he is wearing a jacket with tails but once in the ring it's a short jacket with no tails.
When Barnum goes to see Charity by the beach at the end of the film, they embrace. As the camera shows a close up of their faces Charity's head is on Barnum's right shoulder, but when they reverse the shot to show her expression she is now on his left shoulder.
In the movie Swedish singer Jenny Lind makes sexual advances towards Barnum while they are touring together. When Barnum rejects her advances, she quits the tour out of frustration and starts a rumor that the two are romantically involved by forcibly kissing Barnum on stage after a performance. Barnum and Lind never had an affair. While it is true Lind quit the tour, she did so because she did not like Barnum's relentless marketing of her and decided to tour with new management. The two actually parted on friendly terms.
General Tom Thumb, real name Charles Sherwood Stratton, was only four years old when Barnum hired him. Not twenty-two as the movie claims. Also the movie fails to point out that Stratton and Barnum were actually related (They were distant cousins.).
Jenny Lind reads the opinion piece written by Gordon Bennett, from the New York Herald, where she is flattered by his supposedly printed praise of her: that she "deserves our highest esteem, and most lavish ovations". In reality, Bennett was so lazy, he only changed the headline for this new opinion piece about Barnum. He actually submitted the exact same article, as before, where he had already criticized Barnum as a "purveyor of the offensive and indecent", and only the changed headline makes any mention of Jenny Lind's success, but not by using the words she reads from it.
PT Barnum and his wife Charity have two daughters in this film: daughters Caroline (born in 1833) and Helen (born in 1840). No mention is made of their younger daughters Frances (born in 1842) and Pauline (born in 1846).
At the start of the movie we see young PT and Charity at the sea shore. Barnum was born and raised in Bethel, CT which is a good 22 miles from the Long Island Sound beach in Bridgeport. Also, Long Island is easily visible from the Bridgeport beach but is nowhere to be seen in this movie.
When PT Barnum brings his wife and two daughters to see the mansion he just bought, they are transported to the mansion by a buggy drawn by two zebras. That would be impossible in real life. The zebra cannot be tamed or ridden as a horse can.
While the animals appear to be zebras, they are in fact horses painted to create the illusion of zebras.
At one point during the song "The Other Side", Phillip Carlyle jumps down from the bar, onto a barstool, and then onto the tile floor. You can briefly see a raised square underneath the barstool, covered in the same tile as the floor. This is obviously a platform that the barstool is attached to so that it is stable enough for Carlyle to jump onto without it tipping over.
Around 2/3 of the movie in we see Barnum leaving with Lind to start her US tour. The film shows a horse drawn carriage waiting outside the door, with Lind already seated inside, it collects Barnum, and then leaves down a gravel drive from the house leaving tracks as it goes down the drive. However there is only one set of tracks that is being created as the carriage leaves, there should have been at least one more set as the carriage came to the house to collect Barnum, as well as other marks from use, such as foot/horse prints.