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.
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 Appeals (the state's highest court) are called "Judge".
Throughout the series, the detectives (or the Crime Scene Unit Forensic Technicians) are able to ID a bullet caliber from the wound size. In reality this is impossible. A 9mm, .38, .40 and even a .45 all make wounds that are indistinguishable from each other on a body. The police also often look at a bullet and ID the pistol from it. While possible, this requires forensic analysis and is generally not very conclusive because the bullet is too deformed. The conformation of a particular bullet coming from a particular gun using "ballistic fingerprinting" has never resulted in a conviction.
When the detectives are interviewing someone or working a crime scene, they are never seen taking notes. Real detectives are constantly taking notes. The notes are so important that they are occasionally booked into evidence to ensure the originals will be available for review before trial.
When the Police need to arrest a medical professional, they are frequently shown barging into his/her office, exam room, or even the operating room. This would never happen in real life and is strictly forbidden, as it is a gross violation of the patient's privacy, and in the case of the OR, could contaminate the sterile environment, thereby jeopardizing patient safety as well.
The series makes it seem like the prosecutors and defense attorneys are evenly matched, which is way off. Lots of defendants can't afford a lawyer and are assigned public defenders. Public defenders can be tasked with 500 cases in a year. (And that's a conservative estimate.)
Throughout all the different New York based Law & Order programs, when the police want to obtain a confession - real or coerced, they will often use the possibility of the defendant being given the death penalty, despite the fact that the state of New York hasn't put anyone to death since 1963. However in the episode Aftershock (1996) we see the aftermath of an execution, this means the Law & Order universe does have active death penalties even if the real world does not.
Though it's more common for detectives to have many cases open and working at a time, and may devote a few minutes or hours to several over the course of a single day, having them give 100% of their time and attention to a single case every week is a storytelling practicality; given the show's two-part format, a compression of time is necessary in order to fit everything into 45 minutes.
During a number of episodes which portray trials which extend over a multi-day time periods, the jury is often shown wearing the same clothing as during the beginning of those trials.
Throughout the series, several actors/actresses play several different characters, with a minimal attempt at disguising this. It's become so well known for this series that the actors who do this are called "Repeat Offenders".
In several episodes, the defense attorney warns that his client cannot be charged with the same crime if the district attorney's office drops the charge in question. This is not true. If a jury acquits or convicts a defendant, then he or she cannot be prosecuted, again, for that particular crime. However, a disagreement over plea bargain does not give the defendant protection against being tried, again, once new evidence is presented. Charges, in these cases, can be re-filed.
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.