In Carmilla the Series, humans have been shown repeatedly to be absolutely no match for vampires in a standup fight. Enter Sherman Hollis, Laura's dad, who has formulated a pepper spray that works on vampires. In his first appearance he drives off the vampirized Danny Lawrence, who physically overpowers Carmilla Karnstein as much as Carmilla overpowered her when she was human.
In the Nat One Productions story-line, The Last Stage, each of the main characters is really nothing more than a Red Shirt. Despite this, they successfully tackle a surprising number of lethally dangerous paranormal enemies. That's not to say there aren't casualties...
In The Gamer's Alliance, several heroes and villains have no special or magical powers and yet they've shown their skills many times against stronger opponents.
Several staff members of the SCP Foundation are (with the marked exception of Dr.kain and Dr. Bright) humans without paranormal abilities, but they can take down or capture dangerous abnormal objects, and in some cases kill Reality Warpers.
Tech Infantry is a world with vampires, mages, werewolves, and 8-foot insectoid alien killing machines, and yet Erich von Shrakenberg, Icarus Hicks, James Welthammer, and the entire personnel of Earth Fleet and the Light Infantry manage to hold their own.
Outside of the Red Panda's use of hypnosis, neither he nor his sidekick the Flying Squirrel have any significant superpowers to speak of. Their success as crimefighters and superheroes comes from constant training, cunning minds, and gadgets that allow for feats ranging from Wall Crawling to resisting and countering magical attack. They can reliably handle anything that isn't a flat out Kaiju.
The Red Panda also has a network of agents, men and women who act as his eyes and ears and have fought alongside him both figuratively and literally. In the book "The Android Assassins", several of them are the first line of defense against Captain Clockwork's robot army and, during World War II, many of those same agents formed a unit that gained a reputation for doing impossible things behind enemy lines and, for much of that time, any superhero help they got was either incidental or the acts of an amnesiac Red Panda.
Sensei Tetsuo Ito, martial arts instructor at Whateley Academy in the Whateley Universe. A little old man, he starts every term's aikido classes by picking the most dangerous mutant in the room and demonstrating that he can kick said mutant's ass.
Chief Delarose certainly qualifies, too. Not only is he the head of security at a superhero school (a job which carries its own innumerable worries), but Deathlist, a Hero Killer of the highest sort, knows him at a glance.
Erik Mahren was, prior to his explosion caused by late-term manifestation, referred to as 'The World's Most Evil Baseline', and managed for years to pull off some simply insane tricks with little more than training... such as putting a very powerful power mimic in the infirmary with two dislocations, and managing to be one of the very few surviving members of the Dragonslayers. They, themselves, were a Marine Corps squad of low-level Exemplars and baselines who took down ragers, monsters and other paranormals which had stymied entire superhero teams for years.
There is in-canon speculation (and occasional backhanded confirmation) amongst many that Doctor Diabolik may in fact not be a mutant, imbued, or even dynamorph holder, but just a very, very intelligent man. Even then, he has become a household name and the top of the B-list of villains.
In Darwin's Soldiers, the nonpowered heroes (villains) usually put up a decent fight against the augmented villains (heroes). They sometimes win.
Any analytical player with the Villager role in Comic Fury Werewolf arguably counts. If they catch the Werewolves, they almost certainly do. If they consistently catch the Werewolves, they definitely do.
As such, Ranger used to be one, but later lost his touch.
Marvels RPG has rules for making your character one of these.
The Metaverse distinguishes itself from other superhero universes (outside of being live action) by featuring unpowered heroes as the majority as opposed to the looked down upon minority. That isn't to say that sci-fi in anyway takes a back seat; giant monsters and alternate dimensions are anything but scarce.
The superhero story Worm has quite a few non-powered characters who attain this status, ranging from the mercenaries hired by supervillain Coil to the Parahuman Response Teams run by the U.S. government. The most prominent such character is probably Brockton Bay PRT Director Emily Piggot.
On the villain side, there's Saint, Dragon's nemesis.
"Batman vs. Spiderman". In this case, the Badass Normal loses. Batman has quite famously beaten superhuman opponents before, but this was with extensive research and tailoring his load-out to specially counter it. Under standard Death Battle conditions, fighters go in with no prior knowledge and iconic, standard equipment. Against an opponent who is even stronger, tougher and faster than a peak-condition human, who can also nullify his stealth and is just as clever as he is, Batman simply can't make up the difference in raw power.
"Solid Snake" vs. Sam Fisher". In this case, the Badass Normal loses. Sam has his Navy SEAL and Green Beret training, and he has superior technology and weapons. However, Snake is not only also trained by US SF, but is also cloned from the greatest soldier to have ever lived and has performance-enhancing nanomachines coursing through his veins. Combined with Snake's more... unorthodoxtactics, and he manages to cinch it.
"Hercule Satan vs. Dan Hibiki". In this case, the Badass Normal wins. Hercule is Overshadowed by Awesome by the DBZ cast but can hold his own very well against other humans, while Dan is a joke even among normal humans. Hercule's feats are also far more impressive than Dan's, whose Ki Attacks are inept. With this huge difference in competence, it was only a matter of time before Mr Satan finished off Dan (albeit accidentally)
On Not Always Right, there are a few of these. One of the better ones was a middle aged lady in a pet shop who foiled a robbery/assault on the clerk so quickly that they had to play the security footage in slow-motion to figure out how she did it. And she didn't say a word after, just bought her cat food and walked out.
Detective Michaels in A World Less Visible goes into battle alongside werewolves, witches, demons, and an archangel, and manages to hold his own.
Diath in Dice, Camera, Action! holds his own against vampires and giants despite being the only non-magic party member.
Washington in Red vs. Blue, especially compared to the other Freelancers. He has no A.I. or any armor enhancement but is able to fight on equal grounds with the others.