In season 1, We're told that Jimmy's age is 22 years old and knowing that Jimmy is a war veteran (WW1). He is likely to have enlisted at the beginning of the US Involvement on the Western Front (1917 - 1918); the problem is that Jimmy's age would have been 19 years old, when he enlisted and the US enlistment age was 21 to 45 making Jimmy... 3 years young for enlistment.
Jimmy would have been 19, when he enlisted in ww1 and he is 2 years young for enlistment in the US Army.
Al Capone's son, Sonny, is portrayed as being entirely deaf at the age of two. In real life Sonny contracted an infection that left him only partially deaf and only at age seven. However, this series is intended to be fictional, not based on real life events. Factual inaccuracies are therefore not considered goofs.
Prohibition is in effect, and yet, in at least 2 episodes of season one there is a Budweiser sign to the right of the Lucky Strike sign on the boardwalk. It seems this was corrected in the final episode of the season.
The series never showed most of the numerous large hotel properties that existed in Atlantic City during the 1920s. While this likely due to the cost of doing so even using CGI, it also makes the city look much smaller than it was.
Both Western Electric and Automatic Electric candlestick telephones are shown. These were competing manufacturing companies that would not be seen in the same geographical area. The "Bell System" would use only Western Electric telephones connected to their telephone lines. Independent telephone companies would use Automatic Electric and many other manufacturers telephones connected to their telephone lines.
Throughout the series, diesel trains are often heard in the background. At the time, only steam locomotives were used on mainline railroads.
While much of the clothing was period specific for the major players, numerous minor performers wore clothing that was clearly made in the later 20th and early 21st century. This becomes most obvious in the latter seasons of the series.