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 Gamers 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.
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.
In Darwins 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.