Target X loves to blur the line between Heartwarming and Tearjerker:
X-23's bonding with her 'cousin' and 'aunt' (No genetic relation, but the scientist who created and carried her and basically was her mother was related to them) was extremely sweet. She even cut off her own hand to save them.
There's the fact that X-23 trusts her cousin, Megan, enough to tell her the truth and show how her claws work. Even though it's dangerous, she still does this so that Megan knows the truth about her kidnapping many years ago, and that she didn't imagine the whole thing.
Laura and Megan playing hooky from school after getting suspended (and stealing the principal's car!) adds Crowning Moment of Funny as well, but then you realize that all of what they're doing is just the sort of things Laura was denied as a child. For one afternoon she finally gets to experience what it is to be a normal teenage girl. The scene from the pet store where Laura is practically buried in a pile of puppies and kittens is especially adorable.
Just before Megan and Debbie leave to go into hiding—the last time Laura will see them—Megan gives her a locket containing a picture of herself and Laura's slain mother.
Laura's confrontation with Logan. She goes there fully intending to kill him and then herself, her experiences having convinced her that the only way out is for both of them to die. From the start Logan does nothing but try to talk her down, and only fights back in earnest when he realizes that it's the only way to get through to her. When she defeats him anyway and Laura explains why she's doing this, he reveals that he knows everything about the Facility, and fully accepts responsibility for what she's been through. He then resigns himself to die but pleads with her not to throw her life away: He accepts guilt because he volunteered for Weapon X, but insists that Laura was an innocent victim because she was denied that choice, finally breaking through to her.
The flashback to the carnival in The Killing Dream. Watching Logan try to help Laura feel like a normal teenage girl is sweet enough, but the conversation on the roller coaster when Logan states his intent to adopt her really crowns it.
Logan: I may not be a good father, but I'll be yours.
Laura defying her off-book orders from Zander Rice to kill Martin Sutter and his family, when she chooses to spare his ( actually Rice's) son. After all the effort the Facility went through to break her, that one moment proves her humanity hasn't been completely destroyed. In The Killing Dream this act becomes one of Laura's most important weapons for throwing off the demon attempting to recruit her into his service.
In NYX, who's the first person to calm and comfort Tatiana after her very public manifestation nearly gets her lynched by a mob? Laura.
The Cool Down HugTeen!Scott gives her to calm and comfort her when Laura is in the midst of a Freak Out! after Murderworld and the torture she experienced at the hands of the Purifiers after that, simply because he believed she needed it. Then, in the following issue, Laura initiates one with him after Stryker, Jr. reveals Arcade released the Murderworld footage (meaning the entire world has seen her in a Trigger Scent-induced rage, something Laura is understandably sensitive about), which threatens to send her right back into a Heroic BSOD.
And then Laura returns the hug aboard the Guardians of the Galaxy's ship, when Scott is overwhelmed both by the capture of Jean Grey by the Shi'ar, and the revelation that his father, is alivenote not only does Teen Scott come from a time before he learned that his father survived the plane crash which he believed killed his parents, at this point the entire galaxy believed Corsair was dead.
Laura and Warren's date in All-New X-Men #30. It starts with a wild night at a club (which culminates in a brawl against some thugs who take a disliking to Warren and his wings), but leads to them having a long heart-to-heart talk and Laura thanking him for taking her away from the insanity which normally surrounds her, even remarking that Laura is prettier than Jean Grey when Laura starts expressing her self-doubts about why he likes her. It's a rare moment of Laura getting to forget about everything that's been done to her, and just being a normal girl.