Skiing off a cliff, holding the audience in suspense for several seconds, then opening the parachute. In the colors of the Union Jack. Nowadays BASE jumping is seen everywhere, but when this movie came out, no one saw it coming. There are audiences who stood up and applauded at that. Stuntman Rick Sylvester deserves special mention, since he was the guy who actually traveled to Baffin Island and skied off the mountain.
The sequence in general is well done (except for the shots of Moore against a blue screen and the banana yellow ski suit), including Bond skiing backwards and flipping off a ledge.
According to the documentary, Everything or Nothing, this was the moment that Bond changed from a successful media property that happened to be British, to a national British icon. It was so well done that at one showing Prince Charles got up and applauded!
The opening credits sequence, coming off of that, beautifully underscores one very important fact: even after several stumbles throughout the franchise's history, at the end of the day, Nobody Does It Better than Bond.
Fighting a henchman, who nearly falls off a building to his death but manages to keep his balance by grabbing onto Bond's tie. The henchman reaches out for help, Bond grills him for information. The henchman tells him everything. Then Bond slaps the tie out of his hand and he falls to his death. Bond walks nonchalantly away, saying "What a helpful chap."
The car/motorcycle/helicopter/submarine chase, with the Cool Car to end all Cool Cars. Slightly less serious, but still awesome.
Q: "I want that ready for Akbar's tea party."
The elaborate battle sequence inside the hollowed-out cargo ship, with Stromberg's forces going up against the combined efforts of American, Soviet and British submarine crews. With a lot of nameless sailors making Heroic Sacrifices all over the place.
Made doubly awesome by the fact that no stage in existence at the time was big enough to shoot the sequence - so they sacked up and fucking built one. To this day, despite 2 fires and subsequent rebuilding, it remains Europe's largest stage, and has been used in hundreds of productions besides the Bond films.
Bond catching Jaws with the magnetic crane and dropping him into the Shark Pool. Which is then topped by Jaws killing the shark.
Anya winning the long duel over who gets to keep the microfilm. "With considerable ease, I might add."
Stromberg's Cool Boat freaking opening up and swallowing three submarines. Also a meta awesome moment, since they had to build the biggest sound stage in the world to properly convey the epicness.
Bond disarming a nuclear missile with what's essentially the world's most high stakes game of Operation. It's really amazing how much suspense the sequence generates even though everyone goes in knowing there's no way he can actually fail.