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: 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.
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."