Courage the Cowardly Dog may be a horror-comedy series but that doesn't stop this show from having heartwarming moments... or entire episodes, even. The things you do for love, Courage.
In the "Curse of Shirley" episode, a curse is placed on Eustace that causes a storm cloud to hover over his head, raining on him everywhere he goes. The curse cannot be lifted until he finds generosity in his heart. Of course, given Eustace's personality, it doesn't seem to be going anywhere. Eustace is eventually forced outside on the roof, where he loses his glasses and starts thinking he's seeing monsters, his hygiene declines as fungus grows on his body, and he seems to be generally loosing his sanity. In an effort to get him back to normal, Courage is told the only way to get him to find generosity is to show Eustace who he really is by showing himself his own image in a mirror. When Courage tries to show him the mirror, he accidentally breaks the glass, and improvises by standing in front of the mirror himself, which results in Eustace seeing Courage as himself as a little boy, miserable and crying because his head is burning out in the hot sun and he doesn't have a hat. Eustace visibly feels sorry for him, places his own hat on his head, and the cloud vanishes. Add the fact that it was one of first few episodes which showed Eustace in a positive light.
A lot of the episodes do this, though in most episodes, it's offset by the Nightmare Fuel. Still, one episode with less Nightmare Fuel comes to mind. A cruel rich man dies, and his shadow is left to walk the earth. He eventually comes to the middle of nowhere and starts scaring everybody. When Courage manages to wear the shadow out, they sit and talk. The shadow tells Courage that his true dream is to be a star. So, Courage suggests that he goes and be a shadow for real stars up the sky (as opposed to the Hollywood stars he was referring too). Err... it's more heartwarming in context.
The episode "Perfect". After being hounded by a schoolmarm about being imperfect for the entire episode, Courage is suddenly reminded that there is no such thing as perfect and he's fine just the way he is. Eustace also seems to start to like him and the family is at once all happy together. Seeing as this was the finale to the whole series, it can be noted that it was a pretty nice ending.
What the fish in the bathtub tells him brings it on home. It's basically telling him to be himself:
Fish: There's no such thing as "perfect". You're beautiful as you are, Courage. With all your imperfections, you can do anything.
And immediately afterward, Courage looks around him and sees that the Bagges are perfectly happy with their own faults; Muriel chuckles over some ruined baked goods and Eustace plays his broken trumpet while happily declaring that he should start a band. This is what convinces Courage that the fish was right.
The episode "Remembrance Of Courage Past", which revolves around flashbacks to the day that Courage's parents were taken from him (well, more like "launched into space by a mad veterinarian" wanting to test dog's growth, but...). At the end of the episode, the last flashback is about how Muriel found him and gave him his name. And then, at the very end of the episode, we're treated to the sight of the mad vet being beaten up by the dogs he's been launchinginto space, including Courage's parents.
In the Latin American dub, the scene manages to become even more heartwarming when, upon finding Courage, Muriel says "I'll call you Courage. And you're going to be okay"
A particularly touching episode is the one where Courage is trying to teach a dragon how to fly (so that said dragon won't eat Courage or Muriel... though, he already ate Eustace). The dragon grows increasingly frustrated and chases Courage and Muriel into a lake where they meet a water-dwelling dragon that only eats fish. The heartwarming moment? The first dragon that was unable to fly is a water-dwelling dragon himself (though he doesn't realize this since he was adopted by flying dragons) and is the long-lost brother of the latter dragon. It's a rather touching scene when the first dragon enters the water and realizes, after spending most of the episode lamenting over how he's a dragon that cannot fly, now has a place where he truly belongs.
Zalost and his rat at the very end of "The Tower of Dr. Zalost".
The near-end of "Profile of Courage", when Courage shows the cutouts being paper isn't so bad after all.
With paper-Eustace appreciating Courage.
"Make me too you stupid-I mean, kind dog." He even sounded GENUINE.
All of "The Hunchback of Nowhere".
Most of "Magic Tree of Nowhere".
All of "The Last of the Starmakers".
Courage helping the duck brothers save their third brother from being cooked in "The Duck Brothers".
The understanding of Courage's name. Although he behaves like a coward, you realize that the definition of courage is the quality of mind or spirit that enables a person to face difficulty, danger, or pain. Now think of all Courage has done.
The end of "The Snowman's Revenge" when the Snowman learns that Courage fixed the hole in the Ozone Layer and restored his home, as well as all his friends. He's then reunited with his girlfriend and the two leave in peace to return home. Even more touching as he's one of the few villains to get a happy ending.
The end of "Robot Randy", where Randy finally earns acceptance of his people by making and selling carved reindeer. Especially since he was afraid that they would laugh at him for wanting to whittle reindeer instead of destroy things for fun.
The three-headed chicken was sent to destroy Courage for revenge of killing their father (the chicken from the pilot "Chicken from Outer Space" and then his cooked self returns for "Return of the Chicken from Outer Space"). The mother told them that they have to destroy Courage or else they can't go back home. After failing quite miserably, they break down and cried... but then Courage decided they could 'fake' his demise with pictures, so the three-headed chicken could go back home.