Mine is from a 1961 movie titled “El Cid”. I saw it for the first time over 40 years ago and the fight scene for the city of Calahorra is pretty amazing:
The bank heist/getaway scene from Michael Mann’s Heat.
The elevator fight scene from Captain America: The Winter Soldier
The opening fight scene between Wolverine and the thugs in Logan. Also the fight between Logan/Laura vs the Reavers in the same movie.
Hannibal Lecter’s escape in The Silence of the Lambs.
The scanner duel in Scanners. The famous scene where:
the guy’s head explodes.
Virtually any of the fight scenes in Richard Lester’s Three/Four Musketeers. Every actor was a trained fencer before they filmed them.
Especially the final battle at the convent. Intense, funny, heart-breaking all at the same time. You could tell that Michael York and Christopher Lee were both wiped in the final encounter.
Yes, the Musketeers scenes were great.
The sword fights in the Erroll Flynn version of The Adventures of Robin Hood were good.
Some great combat scenes in the Arthurian epic Excalibur.
For more modern combat, Breaker Morant and Saving Private Ryan stand out.
Neo vs Agent Smith, followed closely by Neo vs Agent Smiths.
Punk’s Hideout Fight Scene in Rumble in the Bronx.
Inigo vs The Man in Black, *The Princess Bride *
The racing scenes in Grand Prix.
For battles, its hard to beat Kurosawa films.
Tyrone Power versus Basil Rathbone in THE MARK OF ZORRO: it’s a story playing out, but each one also seems like he genuinely wants to stab the other guy to death. No, I get that there’s plenty of blade-on-blade parrying, but it never appears to be an end in itself; it always looks like a cheerful killer is maybe just one second away from thrusting a long knife into somebody who’s thinking the same thing.
There is a really good dog-fight scene with Robert Redford, but I forget the name of the movie. 
In the funny fight scene category, I’ll nominate Gamby vs Russell in the final episode of Vice Principals.
I’m not sure it qualifies as an action or fight scene but the battle of wits between The Man in Black and Vizzini in that movie is one of my favorites as well.
Bruce Lee’s battles in Enter The Dragon.
When people accidentally figured out how to use reverb to good effect.
It’s like Bruce Lee and Buster Keaton going through a Rube Goldberg machine. Great choreography and mixing of action and comedy. For those who’d like more examples of this and a deeper look, here’s something you may find interesting, it’s an editor who gives you his takes on movies: https://www.youtube.com/watch?v=Z1PCtIaM_GQ
One of the greatest summits (and pisstakes) of action scenes, a pretty good metaphor for the US military in Vietnam: https://youtu.be/wgzxSr6l9Y4?t=48
The tension, ratcheting up and intensity of this scene are inspiring, humbling and chilling: https://www.youtube.com/watch?v=1csr0dxalpI
This is going to make me look like I’m 12 years old, but…
The early Steven Seagal movies had some really fine one-on-one (or two, or three, or seventeen) fight scenes. As I recall, the stunt guys said that a lot of the contact was real and somewhat painful. It was to the scenes’ advantage that Seagal would totally own his opponents, with each one dispatched in just a few seconds.
I don’t generally care for fight scenes. The fights in Couching Tiger, Hidden Dragon and Lester’s The Three Musketeers at least didn’t bore me.
Bond vs Grant in From Russia with Love. The. Best. EVER!
Kind-of-joking answer: All of Mad Max: Fury Road
Not-joking answer: This segment of Fury Road where the guys on poles come in.
Swordsman 2 (1992) is chock full of mind-blowing action scenes, for instance:
* The Sword of Doom* (1966) has a kick-ass climactic sword fight with a crazed Tatsuya Nakadai taking on his own allies.Chimes at Midnight (1965) is well-known for its masterfully edited - and influential - Battle of Shrewsbury:
