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 Funny Moments 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.
Captain America is dead set on handing Laura over to the authorities, having let her go before and been on the scene at one of her massacres. Even after learning the story he was prepared to go through Matt if he had to. Trying to explain why he had to Steve looks at her, one look is all it takes for him to have her literally Put on a Bus rather than handed over to SHIELD.
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.
Captain America contacted Daredevil explicitly to use him as a Living Lie Detector while interrogating Laura, and only so he can confirm her identity before handing her over to S.H.I.E.L.D. However as Matt listens to her story he comes to realize that Laura is as much a victim as the people she's killed, and actively fights Steve's efforts to take her in. The instant Cap has everything he wants, Matt appoints himself as her lawyer in an effort to protect her from prosecution.
Steve finally recognizing that Matt was right all along and letting Laura go, as well. He knowingly took a tremendous risk turning her loose again, but realized that turning her in to S.H.I.E.L.D. wouldn't have been serving justice, just giving them another weapon.
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's reunion with the NYX kids during her 2010 one-shot. During a time when Laura is still struggling with her identity and her past she finds some comfort that Kiden and the others deeply care for her, and even tried to find her after she left them. Kiden even stands up to Wolverine to protect her when she mistakenly thinks he's one of her old Johns.
Even with the original's return, Laura is still able to use the "Wolverine" moniker, with Logan's support. Demonstrated in this exhange from House of X.
Laura: Actually, I'm Wolverine.
Logan: You tell 'em, kiddo.
The party at the end of issue 5 for Laura and Gabby. Technically it's to celebrate Esme being dead again, but it nonetheless a touching moment between the sisters.