In the Whateley Universe, this is usually an important part of the plot and/or the character. Telekinetic Flying Brick characters like Lancer get flight and anchoring and Nigh-Invulnerability and such from their PK field around (and through) their body. Density changers like Phase, when completely light, also have gravity warping so they don't sink through the floor (Phase can now fly, but when he first got his power, he did have trouble not sinking through the floor). Speedsters like Scrambler and Go-Go are Energizers, need to eat a ton, and do have to worry about things like chafing (Go-Go also has breasts, and has to worry about bounce when she speeds.) Wallflower can go invisible and make things around her invisible: when doing powers testing they found that her 'sight' when invisible wasn't perceiving light, but was psychic.
Frostbite really lacks the Required Secondary Powers. She can move water, but she uses the heat in it to move it, and as the heat gets displaced it all freezes solid wherever it happens to be, and so she ends up being an 'ice manifester' by accident. And she has no protection from the cold she creates. She's given herself lots of colds and one nasty case of pneumonia.
Phase may be an Intangible Man, but he doesn't become intangible as much as he moves matter interdimensionally. So he has no trouble with breathing or keeping his blood inside his circulatory system: it's just that he's moving matter into another dimension. It turns out this means he can go to 'normal density' and 'disintegrate' whatever he is moving through.
It's explained that size warpers aren't actually changing their size, they're putting up warp displacement fields that make them look like they've changed their size, and have all the effects thereof, but they actually haven't. Phase uses this to his advantage once he realises that being a density warper lets him take over the warp displacement fields of others.
Wolverine's required healing factor parodied in thisCracked video.
In the sixth Shadow Hawk story in Epic Tales, Shadow Hawk's required secondary abilities are actually a major plot point, as there is a character who's presence negates them. Also interesting in that Shadow Hawk's powers are magical in nature, while the negation powers are scientific in nature, and result after a section of really bad technobabble.
In the Global Guardians PBEM Universe, it was generally assumed that all characters had these as a matter of course. Exceptions tended to make for interesting side-stories. For example, Silly Putty was a Rubber Man, but he lacked the power to actually control his extended body when stretched; as a result, he had to be manually shaped by other people. The Shield was Nigh Invulnerable, but wasn't anchored (or superhumanly strong), and as a result was regularly knocked ass-over-teakettle when he was hit with any significant force.
It's explained that the Freelancers' AI granted them the Required Secondary Powers needed for individual armor abilities. When Griff tried out the Super Speed armor ability without an AI, everything, including his mental processes and metabolism, was sped up, and he ran straight into a wall because he had no way of knowing when to slow down.
This is actually averted in the PSAUpgrading, when Caboose gets his armor stuck in invisibility mode.
Church: Don't worry Caboose, I'm sure when the game comes out there'll be a way to shut it off.
Caboose: Good. I need sleep.
Sarge: Sleep? When that game comes out, I won't sleep for a week!
Church: Yeah, no, it's not that, it's just that he's having trouble sleeping because he can see through his eyelids now.
Sarge: Oh. That's creepy.
Agent Washington mentions an incident where Agent Utah activated a force field, and almost suffocated to death before he could turn it off. The incident is in a deleted scene.
In Trinton Chronicles it would seem like everyone who has an ability that might normally kill them has an unmentioned secondary power.
Best example of this is Dan who controls time, in order to move at hyper speeds when emulating Super Speed he would need a way to extract air with out burning his lungs and in order to see when he slows time to a crawl he would need a way to speed light up to his eyes while still perceiving slowed time.
The demons in the The Salvation War all have giant back muscles that are very similar to the cells of the electric eels, which makes up for the energy a 20 foot tall man would need. The flying demons are filled with a light and highly flameable gas that allows them to fly and breath fire. However, the byproduct to the gas makes the demon's blood HIGHLY acidic, and while their insides are well protected from the acid, their skin bursts into flames if the blood touches it.
Succubi are capable making themselves invisible to humans, or make themselves look like beautiful women, but they avoid the problems of such powers by only using a form of mind control to make humans see, or not see, what the succubi wants to. However, this does not affect security camera recordings, and can be blocked with a tinfoil hat.
Another example is the healing ability of the angels and daemons. One wounded angel noted he needed a lot more food and water when they were healing, and because of that his was slowed greatly. It is mentioned they have systems in their bodies that prevent such a fast healing rate from becoming cancerous. When an angel army was Nuked, the radiation broke down the systems with horrible results. They, however, also fall to the problem of healing fast does not mean healing right. The broken wings of several angels healed so the bones fused and the joints lost their ability to move.
In Worm, these are handled by the mysterious, extradimensional creatures that grant powers in various ways. In the case of Skitter, who has the power to control and assign individual orders to every insect within a five-block radius, this takes the form of providing processing power that makes her the most efficient multitasker on the planet, while in the case of Sundancer, who creates and manipulates a miniature sun, it makes her immune to flames by shunting all excess heat around her into another universe, effectively normalizing the temperature.
There's also a character who can turn these powers off for her enemies, which can be devastating in the middle of a fight.
Additionally, most of the characters have the secondary powers that prevent their own powers from affecting them. This comes into play with the Manton Effect, which refers to how superpowers can't affect other people in certain ways. You can't Portal Cut into someone's body, you can't ignite a fireball inside someone, you can't use hydrokinesis to pull all the water out of someone's body, etc.
Faultline theorizes that the self-protection powers have also works as a sort of unconscious limiter. Basically, you can't Portal Cut someone else because another person is too psychologically similar to the user. Parahumans who can ignore the Manton Effect are usually much more powerful for being able to do so. It's also part of why Behemoth is so deadly, it can ignore the Manton Effect and spontaneously combust anyone who gets too close to it.
After a major event, one new cape in five now doesn't have those limitations and thus die upon getting superpowers.
It is eventually revealed as to precisely why superpowers have these required secondary protections/ support abilities: they're deliberately included as part of each of the shards released by the Entities to grant powers. Since the shards need their hosts to survive so they can reach maturity while still getting useful applicable data from the hosts and their combat experience, the shards automatically include limitations, secondary abilities, and defenses to protect their hosts from their own powers or allow them to effectively use a power that would otherwise be hard to operate on their own, i.e. Taylor's insect-control.
SCPs with seemingly beneficial properties inevitably have a drawback, often in this vein (such as one that grants immortality but shuts down the a body's ability to heal), that renders them too dangerous to remain uncontained.
The only one which appears to have no side effects whatsoever is SCP-500, a jar of pills that can cure any disease within about two hours. However, there are only 47 pills left, down from 60 when the Foundation first acquired it. And it can't be synthesised.