These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
At the time, jokes about the recent Iranian revolution, the Ayatollah Khomeini, and traditional Islamic dress were acceptable targets. Nowadays, not so much.
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.
"Funny Aneurysm" Moment: An early sketch parodying Question Time featured Rowan Aktinson 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 not long after due to issues relating to the Falklands War.
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.
The show mocked The Two Ronnies, claiming they were old-fashioned and out of touch. Then at the Turn of the Millennium, with The Two Ronnies still being very popular, Ronnie Barker and Ronnie Corbett reunited for The Two Ronnies Sketchbook, in which they replayed some of their favourite sketches and talked about them. The format was then borrowed by...Mel Smith and Griff Rhys Jones of Not the Nine O'Clock News for The Smith and Jones Sketchbook!
That said, although Ronnie Barker was reportedly very offended by the sketch, Ronnie Corbett was very amused by it.
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.
Take That: Aimed at all sorts of media formats. Probably the only one generally regarded as Dude, Not Funny! was when they mocked The Two Ronnies, portraying them as old-fashioned and out-of-touch. (see above)