The Disappointment of the Dionne Quintuplets - S1-E4
Factual error: When Midge is moving to her parents' apartment there is a stack of books on the table while her son is watching TV. On top is a Nancy Drew book (the back of the book which is yellow). However publisher didn't introduce the editions with the yellow back covers until the 1960s.
Continuity mistake: When Midge is doing her makeup routine, she doesn't have eye liner, then she does, then she doesn't.
Factual error: Midge talks about a 2 for 1 pantyhose sale but pantyhose weren't introduced until 1959.
Factual error: The word "floaties "was not used to refer to pool toys in the 1950's.
Factual error: When in Harrisburg, Susie mentions the Civil War Museum, but that didn't open until 2001.
Factual error: Taxicabs appearing in various scenes show that they are fitted with side marker lights on the fenders. Those were not used on American cars until 1968.
Factual error: Abe hilariously exercises in a ridiculous suit while listening to Robert Preston's "Chicken Fat" recording. But Preston (who starred in The Music Man) didn't release this until 1962, at the behest of President Kennedy, who was big on physical fitness.
Suggested correction: It's true that this song was not recorded and released until 1962, which is a little after when this scene takes place. However, Abe is not actually "listening" to it (or any other music) while he is exercising on the lake dock. The song is used as part of the series soundtrack and creates a desired mood for the scene. As the storyline is now in the early 1960s, it is appropriate to the era, and is only background music.
It's Comedy or Cabbage - S3-E5
Continuity mistake: When Midge descends the hotel's grand staircase, the piano player and guests are below. Once down the stairs, Midge finds Abe and Rose sitting in chairs with their luggage, just to the right of the piano, in front of a large pillar, with a round sofa behind. However, when Midge was walking down the stairs, the lobby looks different. The piano is in the same position relative to the stairs, but her parents and the chairs are missing and the round sofa is a different style and the shape of the carpeting in front of the pillar is also different.
Factual error: In the Greenwich Village record shop, the vinyl records are shown with polythene sleeves. At that time, vinyl LP's were always factory sealed in cellophane shrink (unless it was later removed), not poly sleeves.
Continuity mistake: When Midge and Susie go see Michael Kessler, one of the mugs on his desk keeps moving between shots throughout the scene.
The Disappointment of the Dionne Quintuplets - S1-E4
Continuity mistake: After Midge puts Joel's ornamental duck down for the final time before she leaves his new apartment, the position of the duck changes in the following shot.
Continuity mistake: While Midge's dad is questioning her about her new job, Midge starts buttering her toast with some jam. But in the following shot there is no jam on her toast.
Mrs. X at the Gaslight - S1-E6
Continuity mistake: The cookie that Midge trades Noah for a cigarette for keeps moving between shots.
Ethan... Esther... Chaim - S4-E7
Factual error: When Midge and Rose leave the club after the magicians act you can see tactile paving at the crossing. This can't be possible as tactile paving was first developed in Japan by Seiichi Miyake in 1965, so they're 3 years too early.
Midnight at the Concord - S2-E5
Continuity mistake: When Benjamin is driving Midge back from the Catskills to New York City, he briefly is driving on the left-hand side of the two-lane country road (which is not a one-way).
Thank You and Good Night - S1-E8
Deliberate mistake: In the Maisels' 1954 wedding reception flashback, Midge hired professional dancers who perform the "Bottle Dance" (where they slide on their knees). This famous dance, often performed at Jewish weddings and social events, was choreographed by Jerome Robbins for "Fiddler on the Roof," which debuted on Broadway in 1964. It did not exist in 1954 but, as an inside joke, while Midge watches, she says, "That's really good. They should do something with that."