The Reveal of the Secret Test in "Command Crisis", where the team hears a recording of Skipper praising their strengths and talking about how he wanted to make certain they could handle themselves if something ever happened to him.
The ending of "Cute-astrophe". When Alice finds the caged penguins and announces that they'll finally be out of her hair because they're being shipped to Copenhagen, all the zoo patrons protest, forcing her to concede and let them stay.
The penguins getting all teary-eyed over having to let Eggy go, and wondering if he'll remember them. At the end, Eggy repeats Skipper's catch phrase, "Cute and cuddly, boys!"
A blink-and-you-miss-it example from the same episode would be Skipper's expression when Private starts giving his own plan of action.
"I Was a Penguin Zombie". The premise of the episode was that Skipper got a broken wing, and in an attempt to escape the hospital room, triggers a series of events which make Skipper look like a zombie, and the others take the appropriate conclusion and run away with Skipper chasing after them each time he finds them until they start trying to kidnap him (they want to see if they can cure him) in a very painful manner (thanks to the muscle numbing cream severely limiting speech, he can't tell them otherwise). The ending, the effects of the cream finally wearing off and Skipper finally convincing everyone he's not a zombie is what earns it the place on this list.
"An Elephant Never Forgets".
"But he always forgives."
Skipper doesn't want a shot, so he hides. To get Skipper out of trouble, Private volunteers to take his shot. Skipper sees it and then finds out that Private could get really sick (or more!) by taking two shots in the same day...so Skipper bucks up and takes the shot.
The ending of "When the Chips Are Down", especially this image.
Julian going Papa Wolf to save the baby Fusa (a species that usually eats lemurs) once he finds out that it sees him as it's dad. Full details are on the Awesome page.
Operation: Antartica is one where we get Private friendship with Hunter a Leopard Seal (The carnivore species that eat penguins.) they were able to stay this even though the other penguins and the leopard seal scoff at this. With Hunter even saving Skiper, Kowalski and Rico from getting eating and Private saving Hunter dad.
The ending of "Kaboom and Kabust". Kowalski comes up with an idea to calm a more-psychotic-than-usual Rico by trapping him in a bouncy castle used to absorb his behavior, and the plan works! Rico falls out of the bouncy castle, and is too tired to move and gives a couple of exhausted groans before the castle deflates and he falls asleep. The sweetest part is when Skipper picks him up and carries him over his shoulder back to their own HQ.
A moment that's shown in "All King, No Kingdom" and backed up by "The Lost Treasure of the Golden Squirrel" is how much Maurice and Mort mean to Julien, even if he doesn't acknowledge it, to the point where Julien's deepest desire in "The Golden Squirrel" is shown to be his kingdom with Maurice and Mort in it.
Kowalski's one-sided crush on Doris is rather sweet, if a little obsessive at times.