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, Courage shows up hiding inside a box of memory quilts. The Stitch Sisters attempt to throw it away but Courage pops out roaring and pulls out red string while giving the villains not his usual frightened expression, but a look of pure anger, showing how pissed off he is for what they did to his beloved owner. He then quickly ties them into submission and twirls them around into the chair, leaving them to watch helplessly as he saves 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.
Eustace gets one in Katz's Island; 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 chain 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".
In one of the last few episodes, Courage is forced to fight his entire Rogue's Gallery in a dodgeball match... and is getting beaten thoroughly. Muriel gives Courage a pep-talk, telling him that there must be something he can do better than all the bad guys can. Courage promptly takes a deep breath... and screams. He screams so loud and for so long it splits the earth open, causing the villains to fall in. The earth promptly closes in after them, sealing them away. Screaming is an awesome power.
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.
No love for The Mask ? Alright. 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.
Surprised Noone mentioned "Squatting Tiger, Hidden Dog" yet, that episode was basically nothing but awesome sauce, from Courage conquering one of his fears, to him busting down 2 story doors, to going hand to hand in an actual fight(with weapons!) against an evil demonic empress...awesome...
Courage takes a lot of crap from his computer, who is usually helpful, but still snarky as all hell. In at least one episode, 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.