This being a British comedy, some anti-clerical jokes just couldn't be absent, especially during the first season (like the famous "Life of Christ" sketch).
Once Acceptable Targets At the time, jokes about the recent Iranian revolution, the Ayatollah Khomeini, and traditional Islamic dress were acceptable targets. These days, jokes like that would get them in trouble with a lot of special interest groups.
Crowning Music of Awesome: The team were known for their musical pieces, such as "Nice Video, Shame About The Song" (a Take That! at elaborate New Romantic-style pop videos), "All-Out Superpower Confrontation" (a protest song over the global tensions due to the Soviet invasion of Afghanistan) and "Kinda Lingers" (an exercise in Getting Crap Past the Radar). Also often an Ear Worm.
A late sketch parodying Question Time featured Rowan Atkinson playing the terminally boring Foreign Secretary Lord Carrington, who is introduced as "a man tipped to be the next Prime Minister". He had to resign from the cabinet a couple of months later due to issues relating to the Falklands War.
The ABBA parody "Super Duper" was made a very short time before the group's bitter split.
An early sketch parodied the labor negotiations of the late 70's, one condition the employees (Mel Smith and guest Jim Broadbent) demanded was sleeping with Rowan Atkinson's and Chris Langham's wives, as well as Langham's daughter. He objects that she's just 8 years old... In 2007, Langham was arrested for possessing child porn.
A similar one had Rowan, Mel and Griff snatching kids and taking them to a BBC truck for Rolf on Saturday...years later, Rolf Harris would be accused of and later convicted for child sexual abuse.
The (unaired) first episode made a rather unsubtle joke about Jimmy Savile ("Ask him for how much a porter makes"). Enough said (John Lloyd's well-known contempt towards the presenter was more explicitly present in Spitting Image).
The "Stoutist Discrimination" sketch, which used fat people as a stand-in for gay people in debates about gay rights, has become funnier since obesity has become such an issue and some people actually make similar arguments.
There's a sketch parodying the Church of England's objection to Monty Python's Life of Brian by reversing it - the followers of the Monty Python religion objecting to the C of E's "Life of Christ" film which they believe is mocking their holy figure - the comic messiah, John Cleese. "The initials are the same!" complains a devout Pythonist. Quite funny at the time, utterly hilarious now that over-quoting and adulation of Python (particularly by foreigners) has led many former fans to view it as indeed becoming a cult.
Nightmare Fuel: In "Jacques Cousteau's Bath Salts", a fake commercial for the aforesaid product, Pamela Stephenson steps into a bathtub and happily soaps herself only for a tentacle to rear out of it, attack her by wrapping itself around her, and drag her, terrified, struggling and (apparently) naked, underwater until she drowns, whereupon the water turns red. Cue Rowan Atkinson surfacing in a scuba outfit with a bottle of bath salts. Really not clear where they were going with that one.
Retroactive Recognition: Pamela Stephenson was a cast member on this show a few years before she was hired for the tenth season (1984-1985) season of Saturday Night Live, becoming the first female SNL cast member born outside of North America and the (as of 2014) first and only SNL cast member from New Zealand.
Special Effects Failure: At the end of the "McEnroe's Breakfast" sketch, Griff Rhys Jones (as John McEnroe) clutches at his head in the middle of his tantrum and manages to pull off his curly wig. There's a blink and you'll miss it moment at the end of the clip where he realises what's happened.