The bottle of cologne that is shot during the destruction of Riggs' trailer is "Hero." The first time we see Riggs going into his trailer, there is an ad for that same cologne on TV.
In the scene in which Leo is cleaning Martin Riggs' house, you can hear the song "I'm Not Scared", by the short-lived British pop group Eighth Wonder, of which Patsy Kensit was the lead singer.
Leo's "okay-okay-okay" schtick was based on Disneyland employees giving directions to Fantasyland. Originally, Leo was going to be an oily, effeminate character, but Joe Pesci didn't want to play him that way. He pitched the idea of making Leo all-too-eager to please, complete with "okay-okay-okay" ad libs, to Richard Donner. Donner laughed and said, "Do that! Do that!"
The "bomb in the toilet" sequence was used as an early teaser trailer for the movie. The trailer ended with the toilet landing on Murtaugh's car and the voice-over announcer saying "They're not taking any more crap!"
The films' writer, Shane Black, walked away from this sequel after he decided to kill off the character of Martin Riggs. Warner Brothers disagreed with that decision, and he left the project.
One scene that was cut out, but restored in the DVD Director's Cut, is an extended version of Leo trying to show Riggs and Murtaugh where the "house with stilts" is located. In the scene, they are parked off road and Leo is trying to recall the address. He keeps going on and on that the address has to add up to 9 because "nine is my lucky number". Meanwhile, Riggs and Murtaugh look through a map book and randomly pick a street to go down. Following this scene is the one already in the film of them finding the house. The deleted scene further explains Leo's remark "I told you. Nine that's my lucky number," after Murtaugh moans "This is the ninth possibility, Leo."
Throughout several points in the movie, Riggs willfully mispronounces Arjen Rudd's name as "Aryan", willfully calls Pieter Vorstedt "Adolf", and refers to Rudd, Vorstedt, and their associates as the "Master Race". These are all references to Adolf Hitler and the Nazis before and during World War II. "Aryan Race" or "Master Race" was a Nazi ideological form which became a concept for white supremacism. The South African practice of Apartheid at the time was also another ideological form of white supremacy, which is likely why Riggs made these comparisons.
The scene where Murtaugh does his 'Free South Africa' tirade - his statement of 'one man, one vote' did in fact, became part of then-South Africa's president F.W. de Klerk agenda to end apartheid, lift the ban on the African National Congress (in the film, protesters outside the South African Consulate had the ANC flag), and released Nelson Mandela from incarceration. Danny Glover did portray Nelson Mandela in the made-for-cable HBO film Mandela (1987), which was filmed prior to the release of the first Lethal Weapon film.
The trivia items below may give away important plot points.
Riggs intentionally dislocates his shoulder in this film in order to get out of a straitjacket, then has to slam his shoulder back into place. This becomes a running gag throughout the series, where he slams his shoulder back into place after riding a motorcycle off a bridge in Lethal Weapon 3 and Ku (Jet Li's character) dislocates both of Rigg's shoulders in a fight scene in Lethal Weapon 4.
An alternate ending to the movie featured a Thanksgiving dinner at the Murtaugh house which is attended by both Riggs and Rika Van Dan Haas. Director Richard Donner later decided that Rika should be killed to further fuel Riggs' hatred of the South African diplomats. With Rika dead, the Thanksgiving dinner scene had to be scrapped.