One comes early on when Lum saves Ataru from embarrassment by posing as the girl other guys try to set him up with to humiliate him. Once Lum gets him out of there and explains things, Ataru, though highly irritated by the fact that he owes her one, finally realizes that Lum is actually pretty and stops her from flying off by shyly asking her to walk with him for a while. Of course, the Reset Button strikes back in the next chapter/episode, but it still shows that, Comedic Sociopathy aside, these two are the Official Couple for a reason.
Episode 44 has Ataru getting angry with Lum and telling her that she is constantly in the way and ruining things for him, and he tells her to "go home." Which she does... back to her home planet. Ataru then spends the next few days frantically searching for her, and actually cries when he realises she's gone for good. Luckily, Lum simply went to renew her passport to continue living on earth and she left behind a little doll of herself which had a microphone in it, so she hears the whole confession. The next day, she returns, and the look of relief on Ataru's face is utterly adorable. Truly one of my favourite moments of the series.
The scene between Shinobu and Ataru in this same episode is also very heartwarmingly emotional. Ataru has called Shinobu out to the park to flirt and is going through his usual pick-up routine, although he has the Lum doll firmly lodged in his shirt pocket ('over his heart' so to speak). Shinobu simply looks at the doll after he is done being silly and asks if there is something she can do for him. Ataru gets melancholy and says no and thanks her for meeting with him. Its obvious to both of them who Ataru truly loves. However, Shinobu is serious when she asks him if he needs anything and Ataru is truly thankful she came to see him. While any romance between them is now out of the question, they are still good friends and still there for each other.
But the biggest one comes at the very end of the 5th movie, Final Chapter, which culminates into a repeat of the race that caused Lum and Ataru to meet in the first place - except this time, the stake is that if Ataru doesn't catch her, all the memories of the aliens will be erased from the earthlings' minds. But if Ataru says he loves Lum, she'll allow him to catch her. Despite everyone trying to convince him that he can just lie - something Lum herself decides to accept in her mind - Ataru steadfastly refuses to say the words, thinking that if he does, Lum will never know if it's true or not. On the last day, they face off atop a giant mushroom.
Lum: Darling you idiot! Do you want to forget about me?!
Ataru: I'll never... I'll never... I'll never... I'LL NEVER FORGET!!! About you... Lum...
Ataru collapses from exhaustion and drops the horns that Lum lost when she became an adult, which Lum decides to take as confirmation of his affection, despite him not saying the words - she tearfully flies into him and guides his hands to her horns.
Lum blackmails Ataru into going on a date by stealing his address book. Once it's over, she gives the book back and Ataru tries calling up various girls for a date, but finds he "Can't get in the mood."
During a typhoon, the house gets flooded and Lum gets trapped under some debris while underwater. Ataru nearly drowns himself trying to save her. While Lum eventually gets them both out, she gives Ataru the credit.
Any time when Shinobu and Kitsune get together.
The ending of the second episode featuring Ten's mother.
Ten: Mother... the truth is, I'm a bad boy...
Ten's Mother: You, a bad boy? Never.
One earlier episode centered around a giant caterpiller that everyone in class wanted to get rid of but who Ataru insisted on defending (With Lum helping him out). At the end when Ataru was cornered and still refusing to give it up, the bandages he'd wrapped around it suddenly disintegrated revealing the caterpiller had turned into a fairy who thanked Ataru and Lum for protecting her and believing she would grow into something beautiful then flew away.
A later episode features Sakura introducing Ataru to the ghost of a girl who had died after being bedridden for a long time, but before dying she had developed a crush on Ataru after repeatedly seeing him out the window of her house. In order to help her let go of life and pass on, Ataru just had to go on a date with her wearing all the wool clothing she had knitted while she was bedridden. Despite the fact that it was a really hot summer day, Ataru goes along with it with only one moment of protest, and ultimately she leaves for heaven with a smile. The episode ends with Lum teasing that maybe she should become a ghost, and Ataru telling her not to be silly.
Ataru's dream is to have an harem... Yet he passed the chance to get it twice because Lum wasn't in it.
The first time is in the second movie. In exchange for him not interfering, Mujaki offers Ataru to eternally live a dream in which he has an harem including every beautiful girl he knows... Except Lum, that Mujaki didn't want to soil with being involved in such a dream. Cue Ataru refusing to stay in it, because an harem without Lum would be worthless to him, and starting a desperate battle with Mujaki to wake Lum up so they'll be back together.
During Inaba's introductory arc, Ataru found a way to forcibly create a future in which he had an harem. Then he gave a look... And found out that, by getting an harem with all the beautiful girls he knows, he chased away Lum. He destroyed that future.