For a literal example, John Wayne in Big Jakecut a swathe through a gang of desperadoes who kidnapped his grandson. Made even more heartwarming by the fact that Big Jake has never even laid eyes on his grandson.
Practically every one of Jackie Chan's '70s to '80s kung-fu movies (not to mention the similar movies starring other actors) featured an Old Master who looked like a strong gust of wind should knock him over but who invariably mopped the floor with not only rooms full of mooks but also Jackie Chan himself until he learned to respect his elders. Most famously the film Drunken Master, but many others.
Il Duce from The Boondock Saints. He's an undefeatable hitman, who, in one scene, ambushes the three Saints and nearly kills them all. When later the police were investigating the crime scene, the main investigator from FBI decided that the Saints were ambushed by six men, because Il Duce was carrying six guns.
Even though he's gotten older and balder, Die Hard Detective John McClane only gets more badass with age, to the point that he's driving cars into helicopters by the fourth film.
The Lost Boys features a grandpa who only reveals his trope-worthiness at the very end of the film, by crashing his jeep through the wall of his own house and killing the Head Vampire with a hood full of fenceposts. He also delivers the best line, "One thing about living in Santa Carla I never could stomach — all the damn vampires."
How the hell has Théoden not been mentioned yet? He's a 70-year-old manifestation of pure badass.
Yoda is around 800 and nearing the end of his species's natural life span, but can still go toe-to-toe with the likes of Count Dooku and Emperor Palpatine. Even thirty years later, Yoda retains enough power to easily lift an almost-completely submerged X-Wing from a swamp. In the Clone Wars cartoon, he performs a variety of insane feats of the force while taking down a bunch of Giant Robot mooks.
Count Dooku, who not only takes down the respected Obi-Wan Kenobi but the prodigious Anakin Skywalker in the same fight, and still has enough in the tank to go to a fighting retreat with Yoda.
Sith Lord/Senator/Chancellor/Emperor Palpatine. He takes on five of the best swordmasters in the Jedi Order despite maybe 40 odd years lack of practice. He kills all but Mace Windu in roughly 10 seconds, then battles Yoda into submission. Decades later, he's got Luke Skywalker at his mercy.
Old Obi-Wan Kenobi handles himself pretty darned well in a bar fight, and he does go off to deactivate the Death Star's tractor beam alone.
20 years after defeating the Russians alongside the Taliban in Afghanistan, John J Rambo can still single handedly waste the Burmese military, in spite of being a 60 year old man. It helps being thick as a tree and wielding a .50 caliber mounted machine gun.
Hub McCann in Secondhand Lions is an old man who's just suffered from a heart attack and walked out of the hospital as soon as he could stand. He's still able to take on four teenage greasers, then takes them home and gives them his "what every boy needs to know about being a man" speech. He's got a few good lines along the way, too:
After being attacked by a teenager with a knife: "You hold it wrong, son. Not like this. You always do it like this...smooth. (tosses the knife back) Try it again."
His Badass Boast: "I'm Hub McCann. I've fought in two World Wars and countless smaller ones on three continents. I led thousands of men into battle with everything from horses and swords to artillery and tanks. I've seen the headwaters of the Nile, and tribes of natives no white man had ever seen before. I've won and lost a dozen fortunes, killed many men and loved only one woman with a passion a flea like you could never begin to understand. That's who I am. Now, go home, boy!"
Three of the main characters in Kung Fu Hustle (Landlord, Landlady, and the Beast) are shown as unassuming older folk — until they start kicking the hell out of asses.
The old man guarding the Wall in the film version of Stardust apparently got much more badass after having failed to defend the gate nearly two decades ago. When Tristan tries to use the same trick his father used, the old man somersaults over the gate and delivers a few well-executed martial-arts moves. When his father points out that the man is 97 years old, Tristan defends himself by saying that he's had a long time to practice.
While only in his fifties at the time, Principal Joe Clark from Lean on Me, played by none other than Morgan Freeman laying the smackdown on a knife-wielding, teenaged drug dealer by using his megaphone is a badass moment.
Command Sergeant Major Basil Plumley in We Were Soldiers. Hardass extraordinaire, he doesn't so much as duck when the Vietnamese soldiers are taking pot shots at him. He doesn't even bother with a rifle, and uses his pistol for the entire battle. Even more impressive? He's based on a real person.
"Gentlemen! Prepare to defend yourselves!"
"Why no rifle, Sergeant Major?"
"By the time I need one there'll be plenty of 'em laying around on the ground."
Jetfire. His cane is also an axe, he can teleport, he kills two Decepticons single-handedly, and performs a heroic sacrifice so that Optimus has the necessary parts.
Sentinel Prime also. He's been in stasis lock for a couple million years, but can kick ass with the best of 'em. Apparently, being badass is a requirement of being a Prime. He would have killed Optimus Prime if Megatron didn't intervene. Having none other than Mr. Spock as his voice actor didn't hurt, either.
Joe Sarno in Way Of The Gun puts a lampshade on this after getting attitude from some hot-shot younger guys: "The only thing you can assume about a broken down old man...is that he's a survivor." Later, the two main characters shoot it out with some older bag-man, though only Sarno himself proves to be much of a threat.
Although in the comic the specifics of his death are left much more vague, Hollis Mason's death in the extended cut of Watchmen shows that he fits this trope pretty damn perfectly. Although surprised by 6 gang members, he gets in more than a few hits before they overwhelm him. It's one of the more moving scenes in the film.
In the Hong Kong film Purple Storm, the Big Bad Soong is a veteran terrorist who can handle himself well in a firefight despite his head and beard full of white hair.
Don Diego de la Vega (Anthony Hopkins) plays an elderly Zorro in The Mask of Zorro who is still tough enough to trounce his student.
Dr. Loomis in the original Halloween series. This guy took so many ass-kickings from Michael and still came back for more.
In Halloween 5: The Revenge of Michael Myers, where after a decade of chasing the madman, he finally lures Michael in to a trap, then beats him savagely with a 2x4, screaming "DIE! DIE!" for each hit, and not even stopping when Michael was knocked unconcious all the while having a stroke. Payback well deserved!
Donald Pleasance in real life. According to the writer of Halloween 4, he did most of his own stunts in the film. He did all this while pushing 70!
In the Mormon film Passage to Zarahemla, the previously mild-mannered, violin making grandpa scales a wall, breaks into a house, and careens down a staircase with a machine gun in each hand. The guns belong to the two thugs he and a guy recovering from being shot just knocked out. Later he stares down a handful of ancient warriors, armed to the teeth and covered with blood. Very calmly, he puts down one gun, scoops up his granddaughter, and walks away.
RED is built on this trope. RED: Retired Extremely Dangerous.
Grandpa Seth of Troll 2, who, despite being dead, takes on goblins with an axe, makes molotov cocktails, and summons lightning bolts!
Kevin Flynn in TRON: Legacy. (Castor even says "He shut the whole room down just by walking into it!" Or something to that effect.) Doesn't hurt that he's actually God in that world, and proceeded to demonstrate it.
In Breaking The Waves, an old Norwegian man at a wedding breaks a glass with his bare hand to show up a punk kid sitting at his table. He doesn't even have any dialogue.
The Danny Trejo movie Bad Ass. Trejo plays an old guy who becomes an internet sensation after a video is circulated of him being threatened on a bus and kicking the punk's ass. Looks like he spends much of the rest of the movie kicking more ass, chased by Ron Perlman. The tagline is even "He's mean...he's angry...he's old...he is Bad Ass".
The protagonist of Drive Angry breaks out of hell to save his baby granddaughter from a demonic cult. Of course, the character in question is played by Nicholas Cage.
All the main characters in The Expendables are all action heroes at least over 60 and they show they can still do what they do best.
Byron Orlock in Targets; he ends a deranged killer's shooting spree by knocking the punk's gun away with his cane and then slapping him.
The elderly man from Germany in The Avengers, who doesn't take Loki's bull speech about dominating humanity lying down. Word of God is that he's a Holocaust survivor, which makes the scene even more awesome.
Sergeant Mooney from Cellular. It's his last day at the job, and he beats the living shit out of a Corrupt Cop twice his size and half his age. Who was supposed to ambush him, no less. Then he shoots the main villain. Choke on that, Retirony.
Averted in The Wedding Singer as a bar brawl is about to begin, an old man punches the antagonist . . . and it has no effect. He comments apologetically that he was much stronger as a young man.
Godzilla in Godzilla (2014). He's easily the oldest incarnation of Godzilla ever seen—not just in actual chronological age, but in the way he acts. His scars tell the tale of many, many old battles, and at several points in the film, he just looks tired, like an old soldier dragged away from a nap. Several critics even compared him to John McClane.
Charles Xavier, for the films set in the 21st century, he's fairly old. He's at least pushing 90 by 2023.
In the original trilogy, Magneto's somewhere in his 70's or 80's, but he can still kick your ass substantially.
In X-Men: Days of Future Past, as a Holocaust survivor living in 2023, he is, at the very least, in his late eighties/early nineties (X-Men: First Class portrayed him as being around ten in 1944). He still lifts the X-Jet and hurls it at a horde of incoming Sentinels like it's a child's toy, takes a piece of shrapnel to the gut and slightly-less-than-casually pulls it out, and still uses the remaining wreckage to further barricade the path to Kitty and Wolverine.