Dreaming Pearl Courtesan Style is all about fighting with your clothes, namely sleeves, cloaks etc.
Solar Hero Style, the signature Martial Art of the Solar Exalted, can be used with any improvised weapon such as unattuned daiklaves (a type of BFS), stone columns and even other characters.
Dark Messiah style has a similar clause in the list of form weapons. This does lead to an Abyssal being able to beat you senseless with a coffin.
This pales in comparison to the Sidereal charm Generalized Ammunition Technique, which allows anything to be used as ammunition, including abstract concepts such as a shout or someone's fate.
One better: load a shotgun with a Lunar ally transformed into a mouse, fire him at your foe at the speed of sound, and have the Lunar transform into an elephant mid-flight, hitting the enemy with an elephant moving at Mach 2.
The Weapons Of The Gods RPG, based on the comics of the same name features Dugu Four Ultimates: a Kung Fu whose example is parrying your opponent's God Weapon with your tea.
This page lists several home-made Prestige Classes for Dungeons & Dragons that use unusual weapons as part of their gimmick. The Gambler is obviously modeled on the Setzer example listed at the top of the page, while the Necrobounge beats people to death with other dead guys, which is awesome.
By default damn near anything can be used as a weapon in D&D, just not with proficiency (meaning a penalty to your attack roll). Anything blunt can be treated as a club, anything sharp as a dagger or sword, etc. They just won't be as good as actual weapons that were specifically designed to be deadly. One Prestige Class, the Drunken Master, erases these penalties so a bottle or tankard or anything else you happen to have at hand is every bit as deadly as military grade weaponry. And no, there is no rule forbidding you from having such items made Masterwork quality and enchanted.
There is a homebrew base class called the Jester. It cannot gain proficiency with any weapons, but it doesn't have the penalties of wielding something it is not proficient with, or from it being a different size. This troper has personally used a large round table to great effect, using it for total cover, and then using the sharpened legs to stab, and even attached blades to the legs, coupled with metal gauntlets, to roll it through a field of battle, shredding 37 mooks before the table was knocked out of my hands. It was a good day.
One prestige class published in Dragon Magazine, the Branch Dancer, gains proficiency in Tree
The Arms and Equipment Guide has, hidden away in its Adventuring Gear section, rules for using a Butterfly Net in combat. It's basically a melee version of a regular, everyday net - Which arguably qualifies in and of itself as an improbable weapon.
There was a demigoddess of swashbuckling adventure in Second Edition who had the ability to wield anything she could hold in one hand as a +3 weapon. According to her bio, she once killed an army of Tanar'ri using nothing but a stack of books and a potted plant.
In 7th Sea some Swordsman Schools specialize in improvised weapons, specifically Tout Pres and Shield Man (for the Explorers' Society). Such schools are usually not officially sanctioned by the Swordsman's Guild.
In the Pathfinder RPG, based on 3.5 edition Dungeons & Dragons, there is a feat called "Caught Off-Guard". This makes you not only proficient with all improvised melee weapons, but denies your opponents their Dexterity bonus to AC, making them easier to hit. If you're a Rogue, you'd also feasibly get your Sneak Attack damage. That means wackiness like dealing 1d4+ 4d6+ 2 subdual damage with a rubber chicken.
The follow-up feat to Catch Off-Guard is 'Improvised Weapon Mastery', which increases damage and chance to critically hit. In the beta test of Pathfinder, the feat was named 'Razor Sharp Chair Leg'.
An actual feasible build in Mutants & Masterminds, where a simple throwing feat plus Attack Specialisation can let you easily build a superhero with Bullseye-style accuracy throwing whatever you feel like (playing cards, USB drives, computer mice, plaster flying ducks, teaspoons, Warhammer figures...)
In Dungeons & Dragons there is a flaw available only to commoners known as chicken infestation, this flaw lead to a build that gives you a chicken spam attack. however this technically wouldn't work as there is a rule that says logic can over rule other aspects of the game (like using bardic music to aid move silently checks).
Some Prestige Classes in Dungeons & Dragons (such as the Drunken Master) have an ability called Improvised Weapon that lets the uses use pretty much anything he can pick up as a weapon.
In In Nomine, Malakim of Creation have the inherent ability to use ANYTHING as a weapon.
One of the weapons suggested for priests following benevolent death gods in GURPS: Dungeon Fantasy is the "Sacred Shovel of Zombie Beheading".
In Mage: The Awakening, the Adamantine Arrows have a spell entitled "Weaponize Object". It literally works on anything the Mage can pick up, and does damage based on either how tough the object is OR how big it is. Another spell called "Kinetic Blow" makes any weapon that normally inflicts bashing damage instead inflict lethal damage. Combining the two means that your necktie or belt is suddenly a Whip Sword, or that a thrown paperweight is just as effective as a pistol.
Most of the weapons in Magic: The Gathering are serious ones. However, Goblins get to break that rule, even outside of the joke sets. For example, the card Noggin Whack depicts a fish tied to a brick being used as a thrown weapon.
Also worth noting is that while most equipment cards are hand weapons, armor, or other held/worn things, not all creatures depicted have prehensile limbs or even humanoid forms. How exactly is a snake (a regular snake, mind you) supposed to use a battle axe, or a pair of boots?
In Scion, the Epic Strength knack "Titanium Tools" was tailor-made for this trope. Anything the character empowers through this knack while it remains in their hands ignores the effects of the user's Epic Strength allowing for fun feats like beating giants to death with trash can lids, impaling monsters with pool cues, and the like. Epic Strength's rules for throwing things around allow this trope to be applied in ballistics with equally lethal effect.
The Weaponry skill in New World of Darkness games has the "improvised weapon" specialty. In other words, your character can specialize in taking whatever junk is around at the moment and using it to kick ass.
In the short story describing the skill, a woman uses her umbrella to take out her assaulter's eye.
In Hack Master 4th edition, there's a clerical spell called Hurl Animal. It causes any animals within a certain radius of the caster to be propelled at high speeds directly away from the caster's body. Damage when they hit something is left to the GM, but the suggestion is 1d6 for anything smaller than a cow, with proportional increases for larger critters and a minimum of one point of damage per animal.
Funniest when combined with Summon Swarm. Shotgun full of bees, anyone?
TSR's old Marvel Super Heroes game was specifically intended to be able to properly do anything in the comics...including the Fastball Special and variants.
Legend has the Spectacular Beats feat, which allows you to treat anything you can grab during a fight as an improvised weapon. It's essentially designed for modeling a fight with Jackie Chan inside a home depot.
In Prose Descriptive Qualities games, Qualities (or Fortes) are essentially "build your own skills," which means that as long as the Game Master can be talked into it, you can fight with anything you can think of. The rules reward creative thinking and flashy action. If you're not doing this at least once, then there's something wrong.