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 a few episodes of the series the senior detective (usually Lenny Briscoe) will approach a drug dealer whom they know will have information valuable to their case, but the dealer will usually play dumb so Briscoe and his partner will frisk the dealer and find drugs, the cuffs come out and the dealer will spill his guts. This action is called the squeeze and the way it's done is illegal. The proper procedure is that after finding the drugs the police are meant to arrest the dealer and bring him to the D.A.'s office for a plea-for-information deal. As only the D.A.s have the authority to put the squeeze on as they will need proof of evidence in case the dealer's testimony is needed in a court of law.
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.
In one episode of Law & Order, Judge William Wright says that he's entering a not-guilty verdict after the jury foreman reads "guilty". While he can do this using "Judgement Notwithstanding", the procedure is that the defendant's defense attorney must file a JNOV after the trial and in a set time frame.
The detectives sometimes pick up a weapon with a handkerchief or by inserting a pencil in the barrel. In real life, the handkerchief might contaminate possible DNA evidence, and the pencil would destroy microscopic markings inside the barrel, making it difficult to match the weapon to slugs retrieved from a victim's body or a crime scene. Instead, one expert recommends holding a weapon in place with gloved fingertips and sliding a thin, stiff sheet of plastic beneath it.
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".
In several episodes in the first season, Sgt. Greevey shows his credentials to identify himself. However, the badge in the wallet is a NYPD detective's shield which is different from a sergeant's shield.
In real life the same group of police officers working with the same group of prosecutors in one year is highly unlikely. Also the same could be said of the police and prosecutors getting through 22-24 cases per year.
When the detectives want to "bring in" or "pick up" someone for questioning, the subjects are usually located instantly. That might work if they had a consistent schedule they followed faithfully every day, but there are few people who do that. Also, many of the people they are looking for are homeless or otherwise itinerant, and even they don't know where they will be tomorrow.
When witness or suspects are brought to the station and interrogated sometimes it's pretty clear they didn't want to come. If there is no probable cause for an arrest or an active arrest warrant, the police can't make you go anywhere against your will. Once in a while, a wealthy or educated person will assert this, but mostly the cops just walk up to people, put the cuffs on them, and place them in the car.
The time-lines of trials seem to be rushed and take place within days of the crime. Most of the cases on this series are homicides, with a few rapes and kidnappings thrown in. These cases, if not plea bargained, are seldom heard in less than a year after the event.
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.
In-trial pleas or deals would be extremely rare. Once the trial starts, the state would have little incentive not to go for the maximum penalty; the time and resources for the trial have already been allocated.
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.