Shirley says she wants to watch J. Fred Muggs on "The Today Show." Muggs left in 1957, three years before this episode takes place.
Zelda is referring to Shirley as "Mrs. Maisel", while talking to Rose Weissman. Some may infer the address is about Rose, instead.
Joel's mother sees J.Fred Muggs on The Today Show in this episode set in 1960, but J.Fred Muggs left Today in 1957.
When Midge and Joel are sitting at a table in the ballroom of the Phoenician, there's a jazz band doing an instrumental version of "Wave." This song wasn't written by Antonio Carlos Jobim until 1967, several years after this scene is supposed to have taken place.