Frank's monologue about being lonely to Carol in series 3
Frank preventing Liam from being taken into care.
Frank being there for Debbie when Chesney harshly rejects her.
A recent one is the old school music when Carl runs into Frank's arms in slow motion during series seven. At this moment, this troper knew Shameless was back.
Paddy telling an insecure Mimi that he still finds her attractive. This was one of the moments that really humanised the previously terrifying Maguires.
Most any moment between Frank and Debbie in the first few seasons before she became the new Fiona.
Ian had several with Danny, a deaf teenager. My favourite was a scene where they're non-sexually sharing a bed. Ian signs good night, which Danny somewhat morosely returns. Ian does head-to-foot, him taking the foot. After he's laid down, however, Danny gets up, causing Ian, who feels it, confusion. Danny lays down at the foot of the bed. Ian sits up, sees Danny's innocent look, which is truly innocent, just a lonely teenager who needs the comfort of friendship, scoffs in amused fondness, and lays back down.
In Series Seven when Paddy Maguire is talking to a distressed Karen about ten months after Mandy's death. Karen is crying and says "She was my best friend" Paddy looks extremely emotional that someone else is still grieving and replies "She was my little girl."
In the same episode Paddy tells Karen that there's no such thing as in-laws - she's part of his family now. Throughout the episode we see him comforting her and acting as a sort-of surrogate father figure during her mental breakdown.
In the American series:
Sheila going into Mama Bear mode and truly standing up to Eddie once Karen comes home in a crying rage after he publicly humiliated her. She doesn't even realise she's left the house until Frank points it out.
The times when Frank made Dottie laugh, even though she knew what kind of guy he was.
Kevin is taking classes so he can finally learn to read and write properly. He does this so he can write Veronica a love letter.
Another one for Kevin: he was willing to buy out the bar he worked at just to save it from being owned by someone else and being turned into something else entirely.
When Ethel and Maleek are sitting on the bus holding hands.
Sheila and Jodie's devoted efforts to take care of Peggy in her last days. especially considering how horrible she was to them before.
Micky throwing away his gun and getting himself sent back to juvie, if one interprets it as him acting on the recognition of how Ian would be better off without a relationship with a volatile criminal.
Lip's final message to his child, even though it's actually the kid of a character we've never heard of. Jeremy Allen White's performance was just that strong.
Sheila and Jodie taking care of Karen's baby after they steal it from the hospital. Jodie even makes a harness for himself so he can simulate breast feeding the baby.
Fiona embracing Lip after he returns home in the season 2 finale.
Mickey telling Ian he missed him in juvie, especially after he threatened Ian for telling him the same thing the first time we saw him locked up.
The boy at the cancer camp telling Carl he's the best friend he's ever had after he grants his last wish of seeing a real pair of tits.
Debbie's little speech to Sheila: ďIíve seen crazy, and Iíve seen bad for kids. And youíre not either of those things. Youíre nice.Ē
From Emily, Frank, while under post-operative delirium, believes the titular little girl to a younger Fiona, and expresses genuine remorse for what a dick he's been. Emily actually overlooks the fact he's delirious and plays the part of Fiona.
When Bonnie, Carl's new girlfriend is staying at the Gallagher's house, she and Carl are cuddling in his bed. When Carl takes off his shirt, she simply says "I don't like sex", and Carl simply puts his shirt back on and they resume cuddling.
Mickey, on several occasions. But probably the line: "Ian, what you and I have makes me free." Topped off by him literally risking his life coming out to everybody rather than losing Ian. Finally the last scene where Ian and Mickey, bloody and bruised, laugh together.