Fights Like a Normal

Some heroes win the Superpower Lottery. Super Strength, Eye Beams and Energy Blasts make issuing a No-Holds-Barred Beatdown a relatively straightforward matter. So Once Again the Day Is Saved, yay.

Then there's the rest of them. They have bona-fide powers that set them apart from Puny Humans, but for one reason or another, they either can't or won't just rely on them alone. They've taken Boxing Lessons For All Those Other Poor Shmucks, and fight very similarly to a Badass Normal most of the time. This can be for any of several reasons:

The powers don't directly apply to combat, such as Telepathy, Talking to Animals, and X-Ray Vision. However, they may have indirect uses for assisting. Or they do upgrade his natural fighting abilities, but only moderately beyond normal human limits, without anything fantastic such as flight. He has an advantage against the Mooks, but only if he stays on his toes and fights smart.

The character has been Blessed with Suck. His powers can easily wipe the floor with most mooks but are very difficult to use safely, such as having uncontrollable Splash Damage, being Cast from Hit Points, or attracting even more enemies. Or they're very unreliable, either randomly failing to activate (or subject to Magic Misfires) or governed by rules he hasn't figured out yet. In either case, he'd rather use mundane methods than chance it.

The character is Willfully Weak. He's either a Blood Knight who has more fun with Good Old Fisticuffs, or a Smug Snake that can beat you with his powers tied behind his back. Both are more likely to pull out the big guns for a Worthy Opponent.

This trope was also an inadvertent side-effect of Coconut Superpowers in old shows and serials due to lack of the special-effects technology to properly depict fantastic powers.

Compare Boxing Lessons for Superman, where trained fighting is an optional extra the character takes to help with extreme circumstances; and Guile Hero, where he prefers trickery to outright confrontation. This trope can overlap with Badass Abnormal, if the character still needs to mostly rely on previous fighting skills post-upgrade.

NOTE: A De-powered character such as a Henshin Hero only counts if he can voluntarily switch them on and off at will and keeps them shut off most of the time, in which case he's Willfully Weak. The character must have a choice whether or not to (try to) utilize his powers to qualify for this trope. Also, if everyone in his universe has or can potentially learn powers, nobody really has "super powers"—he's just that universe's version of a Handicapped Badass or Willfully Weak Badass Normal.

This trope fought its way into these works the old-fashioned way, because its Wiki Magic was just Too Awesome to Use

Anime and Manga

Alucard from Hellsing is a vampire with supernatural powers. Despite that, he prefers to simply shoot his enemies with custom-made guns rather than use his inhuman abilities, mostly because he considers the majority of his opponents not worthy of using his superpowers against them. (Disregarding the fact that a real "normal" firing those guns would risk arms broken by recoil.)
The titular character of Rune Soldier Louie is a mage, yet he prefers to fight with his hands, using his famous "LOUIE PUNCH!". He is called out on it multiple times.
In Naruto, Might Guy fights almost exclusively with taijutsu, despite being capable of ninjutsu as well.
Li Ho from The Law of Ueki. His power is to turn hair into telescoping bats, but prefers to use taichi moves in combat.
Kamijou Touma from A Certain Magical Index has an Anti-Magic right hand which he can only use defensively, so he usually fights superpowered individuals with Good Old Fisticuffs.
The One Piece character, Bon Kurei can take the guise of other characters. This does make him physically stronger or weaker, but he is most accustomed to fighting in his true form. Admittedly he has been known to sometimes try and pull a Shapeshifter Guilt Trip. Of course, this being One Piece, you might also have to answer the question "what counts as "normal"?".

Comic Books

Black Canary from the DCU may be the single best example. Her only superpower (sonic scream) is rarely used, and she's probably gone on 15-20 issue streaks of not using it at all, but she's a martial artist good enough to give Lady Shiva pause.
Captain America is easily confused as a Badass Normal. But he isn't given that he has a superserum that gives him the ability to be at the peak of human condition.
Combat training is standard for the X-Men, since many of them have powers that are either not specifically combat-oriented or too dangerous to utilize except as a last resort.
Hitman's Tommy Monoghan pretty much stopped using his powers entirely after the first twenty issues or so, getting by on charm, badass, and luck.
Spawn has super powers, but prefers not to use them, since they're granted by hell and each use further damns his soul.
Aquaman used to be like this, before writers began upgrading the crap out of him to counter the whole "LOL, Aquaman" meme. Nowadays he can just summon every shark in the ocean or a mountain sized tidal wave to crush you dead.

Literature

In The Wheel of Time, Rand Al'Thor is the most powerful channeler in the world, but for the first half of the series prefers to use a sword since the male half of the One Power is tainted and he's having serious How Do I Shot Web? and Magic Misfire issues.
In the Warrior Cats series, out of the three cats with superpowers, only Lionblaze's are useful in battle (he can't get hurt in a fight). Jayfeather's power is to enter other cats' dreams (though he doesn't fight anyway, since he's a medicine cat), and Dovewing's power - Super Senses - is actually a hindrance in battle because the amount of noise and scents confuses her.
Richard of the Sword of Truth series has phenomenal wizardly might, but relies almost exclusively on his sword for combat, primarily because he has no bloody clue how to get his magic to work at will. This is a borderline example, since that sword is itself magic.

Western Animation

Princess Sally Acorn was this in Sonic Sat AM, while her intellect and skills with computers were highly useful in missions, her physical abilities were mostly an Informed Attribute, usually having to evade enemies or require Sonic's help when physical combat was required. The Archie comics rendition has more developed combat skills, though still can only take on minor threats head on usually.