Lethal Weapon 2Goofs
Obviously, the subplot involving Leo Getz (Joe Pesci) was about protection for his money laundering against the South Africans. In spite of this, he enters the South African counsulate with Murtaugh, offering a diversion for Riggs to enter. This is making a spectacle of himself in front of the very people he means to avoid.
When Arjen shows Murtaugh's file to Pieter, the DOB states 5-15-45, which would make Murtaugh 44 at the time the film was released. However, Murtaugh was celebrating his 50th birthday in the first film in '86, so assuming the release date is "film canon", his DOB should be in 1936, making him 53. Murtaugh also states that he will retire at 52, which can't possibly make him 53 years old in this movie.
Crew or equipment visible
People with diplomatic immunity do not have automatic carte-blanche to commit any crime without punishment. If a person with diplomatic immunity commits a crime, law enforcement has to contact the host country, then the diplomat either has his immunity revoked for trial or gets deported back to his/her own country to face consequences there. Arjen Rudd would certainly face (at best) life in prison after shooting a cop in cold blood: the South African government would give him up rather than face an international incident.
After the car is driven out of the cargo container and lands in the water with all the money (at the end of the film) As Riggs and Murtaugh run to the end of the hanging container armed with machine pistols, there are already 2 conveniently tied off hanging ropes for our heroes to repel down to the ship.
Incorrectly regarded as goofs
When Murtaugh is showing Riggs the new addition to his home, the contractor fires his pneumatic nail gun. In all scenes, it can clearly be seen that there is no air hose attached to the gun, seemingly making it impossible for it to work. However, some nail guns are powered by batteries, CO2 gas, or even blank cartridges. The nail gun used doesn't appear to have the 'air hose' connection.
When Vorstedt reveals to Riggs that he was the one who mistakenly killed his wife, the whole subplot makes very little sense. If the "contract" was on Riggs' life, it's unlikely that Vorstedt would just drop it after the mistake and fail to proceed to kill Riggs himself. It's also unlikely for a character like Riggs to give up a hot case even after a grave loss. Even if he was pulled off the case, someone else in the PD would have picked it up after him.
The "Liquid Nitrogen" brought in by the bomb squad bring in during the bathroom bomb sequence is just dry ice in water in a Pyrex measuring cup. Real liquid nitrogen has to be kept in a special container to remain a liquid so if they tried pouring it in a room temp open air container it would revert to gas almost instantly and be useless.
When Riggs and Murtaugh discover that most of the cargo inside the container is money, Murtaugh notes that "...they're all $1,000.00 bills." Presuming that the movie is set in 1989, the year of its release (see trivia), all denominations of US currency above $100.00 had been withdrawn from circulation for twenty years. Large denomination bills are now mostly in the hands of collectors and dealers. The huge quantity of such large bills would immediately draw suspicion no matter what their source. Part of the reason for the withdrawal was because the large denominations were not widely used enough; other motives include making it more difficult for criminal traffic in large amounts of currency, just the sort of situation depicted in the film. In 1989, there were approximately 220,000 $1,000 bills still being used, considerably fewer than the number of bills on the pallets.
In the opening pursuit scene, the police unit call signs are given as 20W15 (Riggs & Murtaugh) & 20W12. LAPD unit call signs consist of the Division number, then a unit type letter (W to indicate detectives), then the specific unit or patrol area number. LAPD Division 20 is Olympic Division, which came into existence in 2009, 20 years after the movie.