The earliest episodes use many different numbers for the precinct occupied by the main police characters, before eventually settling on the 27th Precinct (or the "two-seven" as it is usually called) for the rest of the series.
In several episodes, Curtis draws his weapon (a Glock semiautomatic) and we hear the hammer being cocked. This type of gun does not have an external hammer and can only be fired in double action.
In more than one episode, characters refer to the District Attorneys of various New Jersey counties. New Jersey has County Prosecutors, not District Attorneys. However, this error is very common.
Throughout the series members of the New York Supreme Court (which is a trial court) are referred to as "Judge" when in reality they are referred to as "Justice". Only members of the New York Court of Appeal (the state's highest court) are called "Judge".
In several episodes in the first season, Sgt. Greevey shows his credentials to identify himself, however the badge in the wallet is a NYPD "detectives" shield which is different from a Sgt's shield.