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.
The end of the Benizakura Arc features Gintoki and Katsura tearing through an entire fleet of space pirates allied 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. This is made ten times more awesome in the movie; the choreography has to be seen to be appreciated.
In the same arc, after Gintoki had been curb stomped and impaled by Nizo with Benizakura, who saves him? Shinpachi, who jumps down from the bridge and cuts off Nizo's arm.
Kondo gets one in the Yagyu Arc, when, in a rare break from being a comedic stalker, he shows just why he's the Shinsengumi's leader by one-hit KOing Tojo, the leader of the Yagyu top students. And this is after Hijikata, generally a more serious and accomplished fighter, has an extremely difficult time with a less notable student.
Same thing for Shinpachi, who truly shows off his skill at sword-play by beating THE head of the Yagyu clan with ONE THRUST OF HIS SWORD.
The Mitsuba Arc may be very short, but it's not without its awesome moments.
Hijikata challenging all of Mitsuba's evil fiancÚ's minions alone and managing to hold his own despite sustaining heavy injuries.
How does Okita finish off Mitsuba's fiancÚ? By cutting his car, which was going at max speed, in half with a sword! It slices apart around him before continuing past and then exploding behind him.
In the Shinsengumi Rebellion Arc we have Hijikata overcoming the spirit in his sword which has possessed him and turned him into a cowardly otaku. He then uses it to cut down a helicopter.
Gintoki gets one of his best moments in the same arc when he pulls a freaking helicopter to the ground using Bansai's strings.
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 Hosen, former king of the strongest race in the universe, to a stand-still. And then kills Hosen by bringing the sun to Yoshiwara, becoming the town's savior.
Kamui reveals his awesomeness when Hosen 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 Hosen'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.
Just how does Gintoki make his entrance before his second fight with Jiraia in the Red Spider Arc? By appearing from behind the spider web Tsukuyo is tied up in and stabbing Jiraia's hand with his wooden sword, before swatting him away. The fight that follows expertly shows Gintoki's tactical sense as he lures Jiraia into a narrow space where he can't move about before tying their hands together with one of Jiraia's threads and beating him up.
Zenzo earns his in the same arc when he appears out of nowhere to save Hinowa from Jiraia's mooks and takes them all on alone. While the fight is off-screen, the next time he's seen he's wiped the floor with the whole lot of them.
Kagura's and Shinpachi's back to backcurb stomping of Mademoiselle Saigou and Pirako, respectively, in the Four Devas Arc. Especially awesome considering how they're often relegated to back-seat in favor of Gintoki when it comes to the serious fight scenes.
You could see it would happen somehow, but that doesn't make Jirocho and Gin teaming up against Kada's minions any less awesome for that.
In fact, the most badass fight of this arc is the first fight between Gintoki and Joricho, which is also arguably the best rage fight of the serie. Long story short, this is the very first time Gintoki got so mad he does NOT speak a single word and demonstrated how terrifying a berserk he could be. While he did lose this fight, it is justified since his opponent is Joricho, one of the strongest survivors of the previous rebel generation way before Gintoki's generation no less.
The Bagaraki Arc has some, including the battle of sadists between Okita and Nobume, and Hijikata cutting a bullet in half before stabbing Isaburo with the same move.
The Courtesan of a Nation Arc is FILLED with these awesome moments.
The Yorozuya storming through the Royal Palace's army alongside Tsukuyo and Nobume. Some parts of it are especially noteworthy:
Kagura, Nobume and Tsukuyo proving to the guards that the women are the most dangerous among them.
Guard: "Kill every single one of the rebels! No mercy for the women and children!"
Kagura: "Women and children? That's some big talk!"
Nobume: "Naturally, that scum would be served by scum. This nation has no respect for women."
Tsukuyo: "Show that you can stop the women you sneer at. Show that you can defend the castle built upon the tears of those women. Since you've forgotten about them, we'll have to remind you! These women have come to topple a country."
Kagura hitting a freaking cannonball like a baseball with her umbrella and batting it straight back at the Shogun's palace.
Gintoki's reaction to Sadasada cutting off Maizo's arm and mocking him for wanting to see Suzuran again? He goes straight for the kill on the former Shogun and when Oboro interferes he breaks his blade with his teeth and thrusts his sword into his helmet! Never make the White Demon angry.
Things proceed to look bleak after that, though, as Gintoki is poisoned, Tsukuyo and Nobume are having trouble with the Naraku, and Kagura and Shinpachi are about to be overwhelmed by the palace guards. Then The Cavalry arrives in form of the Shinsengumi (Aiding Kagura and Shinpachi) and the Mimawarigumi (Aiding Gintoki, Tsukuyo and Nobume).
This one's a twofer: When it looks like Isaburo has pulled a Cavalry Betrayal and shot Gintoki, Tsukuyo goes absolutely berserk and kills any Naraku in her way to reach Gintoki. As she reaches him, a Naraku sneaks up on her and tries to stab her in the back while she's distracted, only for Gintoki to stab the Naraku through the chest with his wooden sword, revealing that what Isaburo shot him with was an antidote.
The Cavalry then expands when the entire Edo police force led by Matsudaira surround the gates with artillery in retaliation against Sadasada. The real awesome moment behind it though 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 Tendoshu that those who rebelled against Sadasada haven't done anything wrong as they were doing what they felt was right and even proved his point by letting all of his forces point their blades and cannons at him]] to show that it would be wrong to make that a crime, as he would then have to punish his entire army. He was even willing to resign as Shogun to bring Sadasada down with him 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.
The entire final battle in the second movie. Most of major recurring cast is involved in an epic battle in the past against the movie's villain. Everybody gets a moment. Even Yamazaki and Hasegawa.
Mutsu gets in on the awesomeness when she finally shows off what a BadassAction Girl she is by curbstomping an entire ship of Space Pirates and saving Sakamoto. Made even better by The Reveal that she is a freaking Yato!
The otherwise mostly comedic Love Incense arc gets some awesome action at the end when Tsukuyo shoots down cannon shells one-by-one with her kunai... using only one hand.
The arc dealing with the ninjas of Iga is still on-going and is already providing plenty of awesome moments:
Okita vs Kamui in chapter 505. Epic fight that take almost an entire chapter without a single spoken word.
Shigeshige's appearance in chapter 506 after his supposed death. It's the friggin' Shogun indeed!
When Zenzo seems to be on the verge of being killed by the Kiheitai, who shows up to save him? None other than Sacchan!