Although Thomas A. Edison invented the phonograph in 1877, it did not become available for home use until 1896 (10 years after this story took place). In fact the Edison Phonograph Company was not formed until 1887.
The Mousekewitzes boarded the SS Austria, which eventually sank (much like the actual ship did in real life), leading to the separation of Fievel from his family. The film is set in 1885, but the SS Austria, the ship which sank, actually perished in 1858, 27 years before the film takes place.
We see Fievel outside of a mouse school with little mice reciting the Pledge of Allegiance. This wasn't written until 1892, over six-and-a-half years after the 1886 setting of the story. In addition, the phrase "to the flag of the United States of America" is used; the original version of the Pledge featured instead "to my flag", and was not changed until 1923, 37 years after the setting of the story. Finally, as the sound of the Pledge trails off, "under God" can be faintly heard; this was not added until the 1950's.
A mouse on the boat to America tells a story involving a calico cat ("tortoiseshell"). Several times, he refers to the cat as "he" or "him". Unless suffering from an extremely rare genetic defect, all calico/tortoiseshell cats are female.
From Bridget's first appearance, up until after she and Tony kiss, she has a parasol. It is last seen at her side when she sits down with Tony, right before Fievel tells the crowd that there are no cats in America. When Tony and Bridget emerge from the wreckage, it has disappeared; when Bridget leaves for Tammany Hall with Tony and Fievel, she leaves without it.
When Tony pulls Bridget from the wreckage at the market, she has a white petticoat under her dress. When she lifts her skirt up while searching for Fievel during the fire, only the underside of her outer dress is seen.
In the establishing shot of the water tower where Bridget leaves Fievel, the moon is high in the sky. A couple of shots later, as Fievel starts singing "Somewhere Out There", the moon is just rising over the horizon.
During the storm sequence, there are two mice playing checkers below decks. In the first close-up the board is filled with red pieces, most of them crowned. Then, when the boards slides with the waves, the wider shot shows the game almost empty, with only three or four red pieces.
When Tony first sees Bridget, and runs to the hole in the wall, his hat lands on Fievel's head. Although Fievel is still wearing it when Tony falls off the awning, it has disappeared by the time Fievel is trying to pull Tony away from Bridget. Although Tony is hatless in the Tammany Hall scenes, he is suddenly wearing it again when his alarm clock goes off, after the scene at the pier.
At the end of the song "Never Say Never", Henri and Fievel are standing by a hole in the unfinished Statue of Liberty, facing the tablet on her left hand. From inside, the top of the tablet appears to be some distance below the hole, but as Fievel is flown to immigration, the hole is seen level to the bottom half of the tablet.
At the end when the Mousekewitzes look back at the Statue of Liberty we see the front of the statue, and behind it we see Manhattan and Brooklyn. That would have the statue facing south, but the Statue of Liberty points east so if they we're looking at the front of the statue they should be looking at New Jersey.
When the cats are hanging on the anchor, on their way to Hong Kong, Digit says that he needs to start learning how to count in Chinese, and starts counting some coins. However, Digit starts counting in Mandarin, while the majority of speakers in Hong Kong speak Cantonese.
In the opening titles, Fievel's name is spelled "Feivel". "Feivel" is the correct spelling from Yiddish, but due to misspellings on posters, merchandise and the title of the sequel, "Fievel" has come to be accepted as the correct spelling for the character's name.
While in the bottle, a wave washes Feivel up on top of the (yet to be installed) torch, in front of the Statue of Liberty. After he climbs out of the bottle, an wider shot shows that the torch is thirty or forty feet in the air.
The goof item below may give away important plot points.
The main plot of the story revolves around Fievel being separated from his family and then the family trying to reconnect. At the start of the film there are five family members, Mamma and Pappa, Fievel his sister and the baby. To Pappa even makes a sad miscalculation when coming through Ellis Island saying he has five, then corrects and says four family members. Well, after the first half of the film the baby completely disappears from the movie and at the end of the film there are only four family members. Making this more problematic is that the baby returns in the sequel An American Tail: Fievel Goes West.