Some recent TV examples:
Felicity Smoak from Arrow is a pretty egregious Mary Sue (season 2 to present). Once she joins the team, they never have to do detective work or interrogate people, she can just miraculously science/computer her way through almost every problem they encounter from villains of the week. Every handsome, rich billionare superhero wants to be with her. Enemies that challenge the super heroes don't scratch her through luck (a main villain doesn't bother to search her when he takes her hostage, League of Assassins members don't hurt her in fights, she drives off a metahuman by spray-and-pray after he beat Green Arrow, etc.). She has currently inherited a billion dollar empire that she uses seemingly without scrutiny (might change, other characters have done this). She learns how to use a super suit that took the smartest man in the DC universe months to learn in about 30 min. There's other stuff that isn't as big. To the counter, most of her relationships haven't worked out and outside of impeccable luck she's pretty helpless against villains and is physically very weak.

For comparison, a relatively over-powered character in TV that isn't a Mary Sue would be Root/Spoiler from Person of Interest. Despite not having any military training and being a runaway, she's consistently the most dangerous character in a world full of badass spies. She's a genius, both with computers and manipulating people. After a while, something happens that lets her take down huge numbers of military members solo while dual wielding pistols. But she's also completely ruthless, everyone who is her friend doesn't trust her, the people she loves actively reject her because of past encounters, and she needs help from a lot of the main cast. Many consider her crazy and actively insult her. She also spends most of the time she's on screen as a servant rather than a leader and other characters react to her in a way that's consistent with their character.