Elysium: Drakey and Crowe, Kruger's henchmen, who despite lacking exosuits, hold their own against Max and others, with Drakey single-handedly taking out the CCB building and forcing an evacuation.
Ash Williams of the Evil Dead series. This man only wields a Chainsaw for an arm, a double barreled Remington shotgun AKA his BOOMSTICK, a chiseled jaw, and the wit of a total jackass. Spouting off one liners left and right, this man who used to be an S-Mart clerk single handedly takes on an army of Deadites. He goes from a normal guy who is just as easily disturbed about the armies of hell as any other, to one of the most memorable movie badasses in history. Hail to the king baby.
Snake-Eyes from G.I. Joe: The Rise of Cobra (Storm Shadow is his Cobra counterpart), specificly when dealing with the Neo-Vipers. In the Paris chase he manages to do a lot more to try and stop Storm Shadow and Ana than his teammates in Powered Armor. All without having anything special besides a gun and his sword.
S.H.I.E.L.D. agents Phil Coulson, Maria Hill, Clint Barton, aka Hawkeye and Natasha Romanoff, aka Black Widow. They all serve under undisputed Badass Normal Nick Fury, director of S.H.I.E.L.D. And, if Coulson's description is accurate, the whole of SHIELD is an example.
Apart from Captain America himself, every other member of the Howling Commandos is a Badass Normal. Peggy Carter and Col. Chester Phillips from the Strategic Scientific Reserve are certainly no slouches in the badass normal department themselves, either. Special mention to Bucky, who keeps up with him for most of the film despite being normal.
Anton Vanko fits, while he did have a laser-whip harness, he was otherwise unprotected when he was hit by a car, a metal door, and by Iron Man himself. He stayed conscious through all of this.
In Iron Man 3 both Tony and Rhodes show they are capable of fighting supervillains without armor.
Unfortunately, this was averted in the case of Happy Hogan who confronted a supervillain and was immediately put in a coma for his troubles. He gets better in the end though.
Also, the first time Emil Blonsky faced the Hulk, he was just a soldier. He didn't fare well but was seemingly the only survivor against Hulk's rampage and certainly the only one left standing by the end of it. That says something.
Riddick of Pitch Black had this going for him until the sequel retconned him into the last of a fearless warrior race.
Sky High has Ron Wilson, Bus Driver! Until the credits, at least.
Jango Fett from Star Wars Episode II Attack Of The Clones has about the best record in the Star Wars films of a normal fighting the Force-using Jedi. First, he fights Obi-Wan using gizmos and gumption. Jango gets the worse of the combat, but his gadgets last just long enough to allow a quick escape, and Obi-Wan is certainly inconvenienced. Later he manages to kill a distracted Jedi Red Shirt with some blaster sharpshooting. This must have gone to his head, because he ultimately tries to take on Mace Windu and gets rather anticlimatically curbstomped — note that Windu was absolutely elite even among the Jedi.
The Transformers films have an interesting example of how a group of characters grow into the role of Badass Normal. When they start off human soldiers were completely helpless against the Decepticons, as they would shrug off .50 cal machine guns and anything lower with no problem. Lennox and Epps managed to discover (almost by accident) that they were vulnerable to HEAT (High Explosive Anti-Tank) rounds, which they fired from hand-held weapons, as well as the reallyheavy weapons fired from artillery platforms. But due to difficulties mobilizing support crafts and hand-held weaponry not capable of extensive enough damage, they were still relying heavily on Autobot support during the course of ROTF. By DOTM you see soldiers training directly with Autobots in how common infantry soldiers can do real damage against the big robots: using trip wires, blinding them with parachutes or sniper rounds, planting grenades on their feet to disable them... using Badass Normal strategies over heavy firepower. In the final battle of DOTM it effectively becomes Bash Brothers with equal footing with the Autobots and Human Soldiers just decimating the Decepticon ranks.
The trailers for the fourth film show the government hunting down all Transformers, claiming that Autobots are not needed anymore. Naturally, the Decepticons return to show them they're wrong.
In the X-Men films, the average cop or soldier is capable of fighting against mutants. They might not always win but they put up a fight.
The military is Genre Savvy enough to go after Magneto with non-metal weapons and took out quite a few members of his mutant army before the X-Men showed up to lend a hand.
A random security guard was the one eventually took down Mystique.
One cop briefly took Wolverine down with a shot to the head.
William Stryker is as intimidating as Magneto, without any powers!
In X-Men Origins: Wolverine, Stryker was the one responsible for shooting Logan in the head with an adamantium bullet, giving him amnesia.
Shingen Yashida. No mutant powers, no supertech, just a katana. Nonetheless, he holds his own against the Wolverine himself. Oh, and he was weakened by Viper's poison at the time, poison that it appears was intended to kill him.
Harada is a mundane leader of a clan of mundane ninja.
Yukio’s mutant power is a limited form of precognition that allows her to see how people will die. Her strength and skills come entirely from training, yet she kicks about as much ass as Wolverine, a decades old veteran warrior with an indestructible skeleton, metal claws, and an enhanced healing factor.
Wolverine has a very difficult time with the two determined Yakuza on top of the bullet train, who do all the same death-defying stunts as Wolverine without any superpowers at all.
A unusual example of this occurs in Spider-Man 2, where the citizens on the train that Spider-Man just saved see him with his mask off and realize that, despite having super-powers, the "guy behind the mask" is actually just a normal human teenager, who presumably goes to school, and has a normal life like every other teen. Although unacknowledged, it's this fact that makes them all stand up to Doc Ock, to try and protect Spidey. Who just uses his "robotic arms" to move them out the way.
In Underworld: Rise of the Lycans, back when Raze was human, he was strong enough to punch out werewolves. He was eventually turned, removing him from badass normal status.
Godzilla Final Wars has Captain Gordon. He defeats superpowered mutants and aliens armed with laser guns using only his fists and a katana. He was also able to survive a few encounters with Godzilla himself.