Any serious arc has a good number of these. Most involve Gin getting up from the brink of death in order to beat the living hell out of the story arc's antagonist. There are more subtle ones scattered throughout the mostly disconnected comedic episodes as well.
The Gintama movie deserves one for outselling a lot of anime movies out there! Link is right here. Take note, this is the collectors edition beating the likes of the Disappearance movie, and beyond.
If Okita gets serious in his combat, expect awesome. And a great deal of sadism.
Episode 61 featured Gintoki and Katsura tearing through an entire fleet of space pirates employed by Takasugi, before the two of them declare that the next time they cross paths, they'd show just as little mercy. And then they jump off the side of the airship they were on amidst a hell of a lot of explosions.
Made ten times more awesome in the movie. The choreography has to be seen to be appreciated.
For that matter, the entirety of the Itou arc. All of it.
Okita in particular gets one of the most awesome moments in the series so far, when he saves Kondo from a carriage full of traitors before killing every single one of them.
Kondo gets one in the Kyuubei Arc, when, in a rare break from being a comedic stalker, he shows just why he's the Shinsengumi's leader by one-hit K Oing the leader of the Four Heavenly Kings. And this is after Hijikata, generally a more serious and accomplished fighter, has an extremely difficult time with a lesser Heavenly King.
Same thing for Shinpachi. We get to see the one reason WHY he's a master swordsman at his school (if you can call THAT a school). He LITERALLY beats THE head guy with ONE THRUST OF HIS SWORD.
Then comes the Yoshiwara Arc, which takes Kagura's awesomeness to a whole new level when she awakens her Yato instincts, turning what was previously her and Shinpachi's near-death into a Curb-Stomp Battle.
At the end of the arc, Gintoki's moment comes when he manages to fight Housen, former king of the strongest race in the universe, to a stand-still. And then kills Housen by bringing the sun to Yoshiwara, becoming the town's saviour.
Kamui reveals his awesomeness when Housen appears to smash him into the roof, causing a shower of gore to fall into the room. Kamui is then shown sitting a few feet away, meaning he switched himself with a prostitute and escaped from Housen's attack undetected in a fraction of a second. He then fights the King of the Night on equal footing, all the while grinning in glee.
What's more badass than Gintoki? Answer: TWO GINTOKIS!!! Everything with the both of them being there was badass.
After getting beat around a bit for the first half of the Jiraia arc, after Gintoki learns the backstory of the main villain of the arc and how he's intending to use and manipulate his own student, our normally lazy, carefree protagonist opens a can of whoopass. Even lampshaded with the title, There will be Hell to pay when a lazy man gets angry
It could only be rivaled by Katsura and Kondo crotch grab on the two Cats from planet Catnip.
The Ikkiou Keisei Arc is FILLED with these awesome moments.
The Yorozuya storming through the Royal Palace's army with help from Tsukuyo and Nobume.
Gintoki's reaction to Sadasada cutting off Maizou's arm and mocking him for wanting to see Suzuran again? Gintoki's Berserk Button has been triggered and goes for the kill but gets in a one-on-one duel with one of this subordinates but gets the upper hand quick as he breaks his blade with his teeth and thrusts his sword into his helmet! Never make the White Demon angry.
But the moment lasts short as the subordinate is actually Sadasada's Dragon Oboro who has a history with Gintoki in the past, gets the upper hand with his poison needles and martial arts.
Then The Cavalry have arrived with both the Shinsengumi aiding Shinpachi and Kagura while the Mimawarigumi led by the healing Isaboro aiding Gintoki, Tsukuyo, and Nobume.
But that doesn't stop there as The Cavalry expands when the entire Edo police force led by Matsudaira surround the gates with artillery in retaliation against Sadasada. But the real awesome moment was that there was only one person that got the forces united against Sadasada. It was the freaking Shogun Shigeshige himself!!!
Gintoki vs. Oboro Round 2 which ends with Gintoki saying the Pre-Mortem One-Liner "Say hello to Sensei for me." before slamming Oboro onto his wooden sword, impaling him!
Later, the phrase was one again used by Takasugi to Sadasada before killing him in cold blood.
The Shogun himself showed his authority by reasoning with the Tendoshuu to prevent any punishment against those who rebelled against Sadasada haven't done anything wrong as they was doing what they feel right and even proved his point by letting all of the forces point their blades and cannons at him to show that it would be wrong to make that a crime. He even was willing to resign as Shogun to bring Sadasada down with them but the imperial order prevented his resignation and at least Sadasada was arrested for his crimes. For the Butt Monkey he has been for the series, this arc has been a big reminder on why he is the Shogun.