The main character, ScottMcCall has the most goddamn adorable big brown puppy-dog eyes. Add to that his heart-melting smile, and you have to either be A) heartless, or B) MamaMcCall to not succumb to it. Kate Argent, the most heartless character on the show, cannot resist them and says, "you, with the adorable brown eyes(, drop your bike, you're staying for dinner) " and "...because I really do love those adorable brown eyes." Although once Kate finds out Scott's secret, well...if you saw Code Breaker, you'd know.◊ ◊◊ ◊ ◊
Puppy enjoying a ride in the Jeep: ◊ and Puppy Head Tilt: ◊ Not even Jackson is immune - this is basically what Scackson's relationship would be:  Seriously, Scott McCall is sunshine and puppies in human form.
Scott's Muggle Best Friend Stiles is also no slouch in this department with huge, golden, doe eyes ◊ ◊ ◊ ◊ that have drawn more than a few comparisons to bambi. The fact he's the token human and usually the most vulnerable of the pack helps the appeal. When he's possessed by the nogitsune spirit, it actually uses said puppy eyes to it's advantage to con people into believing Stiles is still in control of himself and make them take pity on him.
Despite having more hazel-green eyes, Derek Hale is quite skilled at this.
Isaac; especially when Derek kicks him out of his pack, and after Allison dies and he goes home with her dad. (Mr. A. goes on about how he can compartmentalize his emotions, and Isaac says, "But Ican't!" It would take a meaner person than Mr. A. not to hug him.
Dr. Cameron has big eyes and looks like a sad little kitten quite often, usually when somebody points out she takes things way too personally and that she should act more like a doctor, not a friend, when dealing with patients.
Dr. Wilson has big chocolate eyes. Cynical House, MD sometimes laughs at him for it.
Dr. House isn't exactly adverse to looking like a kicked puppy when someone hurts him in one of his weak spots, either. His initial reaction to his cane snapping in half in "Safe" and his emotional moments with Stacy are good examples.
Cuddy: I'm here if you need me. House: ...I need you.
For a more deliberate (and hilarious) version of this trope, see "Deception", where House needs a lab technician to run twenty-nine tests on one cc of blood. She refuses. House compliments her eyes. She starts to say something sardonic and turns—House, chin on a nearby centrifuge, slight sad smile, eyebrows up, gives her a soulful blue look. She did the tests.
Lt. George from the fourth season of Blackadder can do a particularly heart-rending version. Hugh Laurie has a talent for looking like a kicked puppy.
Despite being a self-described shark and a Magnificent Bastard (at least initially), Nathan Petrelli has one of the most effective, soul-melting puppy-faced looks in the business (case in point: Into Asylum). ...Damn you, Adrian Pasdar.
Sylar manages to pull off this look on occasion, when he's not being completely psycho.
Peter, when he's trying to convince Nathan to go along with one of his ideas.
In one episode Q tries to seduce Janeway by making a puppy appear in her quarters. Janeway initially fawns over it, then realizes what's going on and declares, "It's not going to work, Q." Q appears, picks up the puppy and asks, "How can you resist that face?" then gives her the same expression.
And then the female Q pops in, demands to know what Q is doing with "that dog", and clarifies that she's not talking about the puppy.
Star Trek: Deep Space Nine Major Kira has a pair of absolutely enormous, soulful brown eyes, and as such was also quite good at these, although she used them much more rarely.
We have Bertram Wilberforce Wooster from Jeeves and Wooster. He doesn't even really try, he's just like that. Actually, most of Hugh Laurie's characters are guilty of this. D'AWWW.◊
Part of Willow's Woobie appeal is her ability to do this. Must be something about Alyson Hannigan. As Joss Whedon put it — "There is nothing more painful in the world than Aly when she makes her big eyes."
Angel. At least one review of the show placed at least part of the shows success on David Boreanaz's ability to always look like a slapped puppy.
Ted in "Crazy Eyes" (also known as "Swarley"), where he reveals he has a tailored look for impressing women: Head down, eyes looking up, hands in pocket, shoulders up, head slightly tilted, conveying shy vulnerability. He teaches is to Marshall who unfortunately doesn't quite get it right.
Robin has her look, but she doesn't use it that often. It goes like this: biting her lower lip, shyly looking away and thrusting her chest. It's a combination of sad eyes and Coy, Girlish Flirt Pose.
Michael Westen from Burn Notice. Accomplished spy, brilliant fighter, master of the puppy eyes. In the pilot episode he was talking with an old friend Lucy for some help (helping a burned spy would not be good for her reputation). As she continued to say "No" he just kept staring at her, eventually making himself look like a puppy putting his paws on the table. Lucy eventually caved.
Djaq (Anjali Jay) and Much (Sam Troughton) from the BBC's Robin Hood were the undisputed masters of the Puppy Dog Eyes. All it took was for either one of them to look sad.
iCarly: Freddie cannot resist when Carly follows up a request, or asking for a favour with a pouting "Please, for me?"
Occasionally Richard Hammond. Might not be deliberate: his eyes are naturally large and he quite literally looks up to the very tall Jeremy Clarkson. Clarkson likes to lampshade it by pointing out Hammond's resemblance to "that cat in Shrek 2."
Clarkson gets in an epic puppy-dog mope of his own in the S12E6 news segment when he has lost his voice and both his fellow presenters and the audience are merciless.
Tim McGee on NCIS, more so in the first few seasons, was prone to looking like a lost little pup more than a federal agent. Fan fiction writers have picked up on this and several stories simple revolve around different characters receiving the puppy eyes.
Clara's big brown eyes were set in a more-or-less permanent state of this, to the point where it annoys/disturbs the Doctor:
The Doctor: Stop it with the eyes! Don't do that with the eyes, how do you do that anyway? It's like they inflate.
Clara's puppy dog eyes are set to stun in her final episode, "Hell Bent".
Annie in Community uses this often to manipulate the group. Jeff explains this look as her "Disney Face" when she uses it to try to keep the group from being mad at her when she does something that might have made them have to retake a class.
Abed has pretty expressive puppy eyes for a character so often compared to an aloof robot.
Troy: (having just lost at basketball) Donít gloat, itís impossible to guard you, your eyes are too gentle and mysterious.
Kurt Hummel (Chris Colfer) can make grown men cry with a mere look.
Blaine Anderson (Darren Criss) has puppy eyes that can make you coo more than if you were watching bunnies and kittens cuddle.
And Emma Pillsbury (Jayma Mays), which led to a funny scene where a drunk Kurt confused her with Bambi.
The Inspector Lynley Mysteries: DS Barbara Havers, whose large, expressive sage green eyes are more than capable of this, and clue the audience in to her vulnerability when she's covering it up with a scathing remark. It is, in fact, largely because of her eyes that Barbara is such a sympathetic character; if her eyes didn't provide a window into just what a good reason she has to be defensive and prickly, she would come off as a complete bitch in early episodes.
The Wire: Ziggy. Take a look at any scenes where he's upset, but particularly scenes involving Frank, his neglectful and rather disdainful father. For all Ziggy's faults, the jailhouse scene dialogue coupled with his big sad browns is a reason, why some viewers finally warmed up to him.
M*A*S*H: Col. Potter pleads with Hawkeye and B.J. to make Maj. Burns feel like part of the gang, and the two consent to it. Hawkeye says "How can I resist those big Colonel eyes?"
The Woobieness of Niles Crane in Frasier at least partly owes itself to the masterful skills of David Hyde Pierce's huge, sad, brown eyes. He does pull off that look with great success (occasionally earning an exasperated "Don't look at me like that!" from Frasier).
Josh Lyman of The West Wing absolutely owned this trope, particularly the facial expressions that go along with the eyes. He was outmatched only by the even more heartbreaking puppy dog eyes of his assistant, Donna Moss. Josh once explicitly told Donna that he couldn't bear to tell her a particular piece of bad news because her sad eyes haunted him.
Ser Jorah Mormont. Every time he gets "friendzoned", has a rival for Daenerys' affections, and when he was banished.
In Emily Owens, M.D., a blind patient tries to convince Emily to treat his dog. She tries to resist, telling herself that all dogs have sad eyes and that she mustn't look, but she does look and she's in. She then gets her colleague and friend Will to help her. She talks about those sad eyes, and Will agrees — Emily's look did it for him.
In Sex and the City, Charlotte York uses her vulnerable look on her fiancé Tray when they pick up their china, and his obnoxious mother is with them and tries to have her way.
In Psych, nearly all of the characters (with the exception of Henry Spencer) were capable of this trope. Most notably, Gus◊, who is also a sympathetic cryer.
Regina Mills of Once Upon a Time may be the Evil Queen with buckets of bad, wrong, and downright evil deeds to her name, but you try staying angry at her when Lana Parrilla is looking out of the screen with enormous, wounded, tear-filled eyes. Fortunately, shegetsbetter.
On Black Sails, Calico Jack insists that a prostitute in a brothel he's currently running isn't paying rent. After outlying how ruthless and unfeeling he is, she turns on the Puppy-Dog Eyes (amidst helpless sobbing) and he recoils at how powerful they are.
Person of Interest. Subverted when John Reese puts his on, as he's usually being sardonic. Played straight with Bear, as he's an actual dog.
Just Shoot Me!. Maya tries this on Dennis. It works, but not for the usual reason - seeing Maya try to be cute is just so horrific to him that he'll do anything to make it stop.