The titular school has a residence set aside for students whose powers make them dangers to themselves or others, such as the most powerful psychic in the world — who looks like something Lovecraft would have thought up during an absinthe bender. Or maybe the kids who constantly emit toxins, or Love Potions, or radiation. Or how about the cute little girl who has so much Super Strength that she could probably juggle the Hulk — but can't control it at all, meaning a casual hug could destroy steel. And that's assuming they even survive long enough to have their powers start to suck — "Burnout," fatal or near fatal side effects of superpower use, is common enough that it's a constant threat even for fully grown, trained mutants.
There's also the societal drawbacks — the Anti-Mutant version of the Ku Klux Klan in the Whateley Universe are Villains With Good Publicity, and while there's no internment camps for mutants (yet) they are required to register with the government — did we mention the government body that handles mutant registration is corrupt, has been infiltrated by the aforementioned KKK-alike types, and generally not a nice group of people to be around? There are other problems, as well — mutants tend to have "tells" such as really weird eye color (violet, red, etc) which means even the most normal looking of them can be found out. Ultimately, even if they look normal, have no side effects, and all their Required Secondary Powers... mutants rarely are allowed to have a normal life.
Craig on the The Allen And Craig Show has the uncanny ability to throw up whenever he sees or talks to a girl. He attempts to supress it, but ultimately it rules his life.
According to Word of God, Anni of Coyle Command has such amazing abilities as...Poor Eyesight and Extra Pain Sensetivity.
Flander's Company: A French webseries with lots of peopled blessed with suck: A man working at the NASA, who was cleaning a room when a jet blew above his head, who got cybernetics limbs, which enable him to... Move at slow motion. An other, who has the ability to censor the eyes of people in real life like it is done for many videos on the internet, and another who can, while dancing tecktonik, make others follow his moves.
In the web-serial Ash And Cinders Cinder and Azoc can't catch a break. They just go from one horrible situation to the next. From their mother straight up being used for firewood, to an Wicked Stepmother, to various creatures across the land that want to kill them.
Azoc gets it even worse. Not only does he endure all that, but once he gets the Rot, he can't touch anyone directly without burning them from the inside out.
Doodieman. Just... Doodieman. Seriously, a guy who has a superpower revolving entirely around his fantastic ability to crap out amazing amounts of crap. His turds can leave his anus with enough momentum that he overcomes the force of gravity and flies. He stops robbers by aiming and firing small turds, keeps a guy from shooting by burying him with a giant turd, shoots turds with enough force to keep a car from falling off a cliff, and stops up a dam. Not everyone he saves is pleased with this.
Tez from Unexpected Diversions. He's an elf that is invisible to the God of Death, who can niether see him nor seems aware that he exists, despite being the oldest living thing on the planet by about 350,000 years. Which sounds great except when he goes mad from boredom, or is injured or tortured horribly, and the universe seems warp reality to the point where he can't even commit suicide.
Dr. Horrible's henchman Moist has the amazing power to, well, make things moist. His prequel comic reveals it was caused by a plutonium-powered humidifier accidentally making him, essentially, permanently extremely sweaty.
Phelous, resident Meta Guy at That Guy with the Glasses is unkillable. That is to say, he dies all the time, but comes back. The downside? He dies all the time. And comes back...to watch crappy horror movies which have left him pretty deranged.
Brendon McKellar in The End receives visions of every nasty break-up and death of a loved one he will ever experience and can't do anything to change the events he sees.
In the world of the SCP Foundation, manifesting any kind of paranormal trait, no matter how innocuous, is a one-way ticket to spending the rest of your life locked up in a Foundation cell if you are discovered (and you will be discovered). SCP-1364 probably has the worst "power" imaginable: super-vulnerability. Everything hurts it severely, even bright light. It can't even eat or drink without hurting itself (fortunately it doesn't actually need food or water). Most SCPs have containment procedures meant to protect the world from them; 1364's are entirely to protect it from the world.
In this web skit, two slackers go to a doctor to get Wolverine's mutant powers. The problem here is that as the doctor explains it, "Wolverine" is TWO mutant powers, adamantium claws, and healing factor. Being the slackers they are, they pick adamantium claws. Unfortunately, the doctor neglected to mention that the only reason that Wolverine can survive with adamantium claws is because of his healing factor, something these guys find out the hard way.
In Chapter 19.2 of Worm, we meet Scapegoat, a hero with a power that allows him to manipulate people's injuries — sorta-removing them from his target and gaining them himself. Oh, and he doesn't get a chance to pick and choose, or even an explicit warning of which injuries he's picking up.
Grace: Problem, S.g.? Scapegoat [rasping]: Hate my power, hate my power, hate it, hate it, hate it.
More generally, about nineteen in twenty powers in Worm fall under this trope, either in the form of an actual drawback that makes it less useful (Rachel's ability to "read" dogs displacing her normal body-language processing so she can no longer read human expressions, or Scapegoat above), or the psychological trauma that triggered the power lends itself to destructive or self-destructive behavior (Lung's ability to gradually turn into godzilla, but only by continually being beaten on, Regent's mind-control power reinforcing his already prominent sociopathy).
The trope is actually pointed out several times, in that gaining powers naturally produces far more villains than heroes. Since Worm's universe has a pretty active and competent government that took special interest in the phenomenon within minutes, being a villain is _not_ a good career choice.
Of particular note is Canary, a singer whose power makes people who hear her song malleable to suggestion. An accidental use of it (turns out telling a jerk to go fuck himself when they're suggestible isn't a good idea) leads to her getting thrown into a maximum security prison for villains, unable to even speak in her defence at trial out of fear her voice would mind-control the court.
In Strip Search, the winner of the competitive challenge has to pick who goes up for elimination that night. While in many reality shows, this would be the best thing to happen, the artists dread this, since they all became fast friends, and don't want to send their friends home.
Using Real Money Trade. Characters seen doing it get their avatar permanently banned so soon after that they don't even get to enjoy their purchase.
The Ashentäk Oracle had the biggest future predicting ability in a city where everyone had this capacity to an extent. She ended up becoming The Hermit and going to live in the local Bubblegloop Swamp because her co-citizens kept harassing her.
Töne Förk. Born with an extremely rare predisposition to be a magical Master of All and rightful heir to the Coalition leadership. However, that meant that everyone who wanted that leadership for themselves had an interest in killing him or making him into a Puppet King. A Face–Heel Turn quickly started looking like the Lesser of Two Evils. The side to which he turned happened to have the way of making him an extreme case of Magically Inept Fighter and he gladly underwent the procedure.
This is a Running Gag on Villain Source (Your Online Source For Everything Evil) which sells awesome superpowers then lists all the unfortunate side effects in the small print.
The premise of Yahtzee and Gabriel's weekly videos, Let's Drown Out, is that they each take turns to play a relatively boring to watch games and after they ran out of interesting thing to say about said games, they got through a list of topics podcast style. Sometimes they picked games that are awesome to watch but they added boring stipulations to keep it in the trend, though half the time it failed miserably.
The characters of Re Evolution experience this with their mutations, which are either useful but have awful drawbacks, such as Dawn Fraser's ability to freeze everything she touches (including her clothes and the food she eats), or are interesting but more or less useless, such as Katie Sharpe's ability to produce pheromones.
The Thrice-Cursed, featured in Polyhistor Academy, are born with innate magical powers, at the cost of a host of negative symptoms.
In Enter The Farside, Farborn and Fartouched don't get to choose what powers they have. Most powers seem to be singular powers with minimal Required Secondary Powers tacked on. Jolly and his crew are the best example. He's 9 foot tall and made of muscle, and just as strong and as tough as you'd imagine he'd be. The downsides are that he's extremely heavy, his skin is green and his hair looks like moss. Some of his crew have it even worse, but all of them are obviously changed by the Farside.
Robynne Darling of Magical Girl Policy has a very powerful "empathokinetic" sense. The blessing? Her ability to read emotions and auras gives her a solid edge against opponents and renders her magically-enforced Blinding Bangs irrelevant. The suck? She has to deal with a high level of psychic and emotional "background noise" (pure torture for an introvert) and has to constantly guard against others' emotions and auras invading her mind and affecting her behavior. (Her cutesy name is a sucky blessing of a different sort.)