It might qualify as a Too Dumb to Live moment, but Eustace back-sassing the Spirit of the Harvest Moon in "House of Discontent". ("At least I got arms and legs! You don't even have a neck!")
Courage defeating the rats in The Nutcracker while dancing to The Trepak.
In "The Quilt Club", after Muriel is sewed into the living quilt, the Stitch Sisters attempt to throw it away her box of memory quilts, but Courage suddenly pops out of it roaring at them with red string in hand. He then quickly ties them into submission and twirls them around into the chair, leaving them to watch helplessly as he saves Muriel. What makes this example stand out is that unlike most times where he's scared but stands strong against his adversaries, here he is genuinely pissed off by what they have done to Muriel. A prime example of why you should never try to harm this dog's family.
In "Robot Randy", Randy traps Eustace and Muriel in a hole in the ground, covered by a stone block. However, he agrees to free them if Courage can beat him in a contest. A break-dancing contest.
In "Stormy Weather," a storm Goddess arrives at the farm searching for her lost dog. She mistakes Courage for her dog and tries to take him away, but Muriel isn't have any of it and argues with the Goddess about it for the entire episode. May not seem awesome on it's own, but this entire fight is going on while the Goddess has created a massive thunderstorm that completely destroys the house. Muriel spends most of it flying through the air, with only some string tangled around her boot keeping her from certain death and the longer this goes on, the worse the storm gets. Does that weaken Muriel's resolve or even phase her? Fuck no! Courage is her dog and nothing and no one is taking him from her. That is real love and determination right there.
Another similar example in "Muriel Meets Her Match," after being framed for a crime she didn't commit and have all her documents stolen, she proceeds to accidentally single-handedly save the day.
Muriel: They can take my old photos, they can take my identity. But when they take my Courage, they've gone to far!
Eustace gets one in Klub Katz; after the Bagges are stranded on the titular island Eustace and Muriel are turned into a wrecking ball and a washing machine respectively and forced to fight in an arena. As Courage (who willingly turned himself into a helicopter) rescues Muriel, Eustace suddenly notices that Katz is in his chair. He promptly flies into a rage and chases Katz around the island with the intent to kill him. Katz was never shown so utterly terrified of someone or something before, but considering what he's like, he totally deserved it.
The fact that not even a tidal wave destroying the ship he's on separates him from his beloved chair and that he goes flying while still sitting in it and not giving a single fuck definitely counts.
Eustace riding an asteroid in "Mission to the Sun".
It's a little understated, but the fact that Courage and Muriel were both on the same page in "The Last of the Starmakers" should count as awesome.
Courage and Eustace's battle in Paris while they're kangaroo monsters in The Transplant.
"The Mask": Courage rescues Bunny from her abusive boyfriend and beats up two of his cronies that were three times his size. When her boyfriend gave chase after her in his car, Courage attempts to pull a Heroic Sacrifice by jumping through the back of his car, taking the steering wheel and driving it onto the train tracks. Also doubles as a Heartwarming moment, as he was saving Bunny for her friend Kitty, who hated Courage because he was a dog. And Courage did this despite the fact that (as well see in his Imagine Spots throughout the journey) he still thought Kitty was a literal monster (despite seeing her unmask).
Courage takes a lot of crap from his computer, who is usually helpful, but still snarky as all hell. In at least one episode, "The Transplant", Courage gets so fed up with the computer's jokes that he rips the monitor out of the wall and throws it out the window.
Computer: Some people can't take a joke...
And, in "Mega Muriel the Magnificent", Courage outright turns it off when it keeps spouting shit.
Eustace gets one in "Katz Candy" for frightening that evil jelly monster away and fighting against Katz over a candy making machine.
"Squatting Tiger, Hidden Dog", Courage goes out to save Muriel again. This time he fights a powerful, evil empress (and the Chinese hipster Di Lung), and goes on an epic quest of trials that could fit in 11 minutes: saving a girl from being eaten by a tiger, narrowly escaping a barrage of poison arrows, catching and spitting back dragon fire with his mouth, and then fighting the evil empress. I repeat. Courage. Was FIGHTING. The villain of the day! Usually he would get lucky or spend more time running for his life, but this time he stood his ground, made a fighting pose and cried "Hi-yah!" The animation was noticeably sped up through the short sequence, and even when Di Lung shackled him, he still managed to reflect the empress's laser eyes back at her via convenient mirror. He took all the lessons he learned (finding his center and using the enemy's force against them) and saved not just Muriel, but apparently all of China. "Yay!" indeed, Courage.