There are some characters who like to walk around with their weapons out in the open, presumably to intimidate other people, or to have it ready right away in the case of a monster fight. However, upon close inspection, you might find that the weapon barely has any signs of being "broken in" - no scratches, no gunpowder...it still looks as sharp as the day the guy bought it.
Some reasons for this may be that the weapon in question has a sentimental value to its holder, can only be used for certain situations, or may just be Too Awesome to Use. At times, it may make one wonder if they might just be better off leaving the damn thing at home.
Subtrope of Useless Accessory.
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Anime & Manga
Kouga in InuYasha carried a katana for ornamentation, and used it only once in the whole story.
Naraku's dragon, Byakuya of the Dreams carries around a sword that doesn't get used until the final arc of the series, where it's revealed that it can absorb any demonic aura within the area and use its power.
A big shiny sword in Zero no Tsukaima turned out to be ornamental and pretty useless. The rusted one however...
Princess Millerna from Escaflowne movie carried a sword on her belt throughout the film, but she never, ever takes part in a single action scene, and never gets to draw it from its sheath, cementing her status as a Faux Action Girl.
In Toriko, Starjun carries a sword with him, but he's never seen using it.
Astérix has a Celtic sword, but fights exclusively with his fists. He does use it for Flynning twice, and once for cutting cake, but he's never allowed to hurt anyone with it on the grounds that it would be too violent.
Power Forged weapons in the Wheel of Time look like this because they have been enchanted to never break, rust or need sharpening. One character mentions that over the thousands of years since his sword had been made his family has only had to replace the hilt.
In Monstrous Regiment Maledict carries a sword he doesn't know how to use. As a vampire, he doesn't need one, but people see the sword and don't attack him.
Almost everyone in Warhammer 40,000 has one, such as Chaos sorcerer staffs, Tau bonding knives...
Many weapons in Second Life don't actually do anything unless they are specifically scripted. While scripted weapons are available, unless you are in a roleplay or combat sim most swords, guns, hammers et cetera are only for decoration.
Angeal Hewley from Crisis Core carries a gigantic Buster Sword around him on missions, but only maybe uses it a handful of times. Zack even comments on it during the mission in Fort Tamblin (dialogue not exact):
Zack: I've never seen you actually use that thing. Don't you think it's sort of a waste?
Angeal: Use brings about wear, tear, and rust — and that's a real waste.
Of course the other half of the reason is that the Buster Sword "is heavy and unwieldy", as an extra scene (in Zack's DMW) reveals. Crisis Core actually deconstructs some iconic elements from the sub-series for laughs like that…
Not anymore; the new ending to Advent Children Complete reveals that Cloud polished and repaired it, and used it as a memorial in Aerith's church, overlooking Aerith's Lake (that contains the Great Gospel).
Captain Falcon and Snake have guns on their person in Brawl if you look closely, but naturally they don't get any use either. Sheik also carries a short sword or long dagger on her person but it likewise isn't used.
Ryu Hayabusa's sword in the Dead or Alive series. Same goes for Hayate's and Kasumi's swords.
In Dragon Quest Swords: The Masked Queen and the Tower of Mirrors, your partners carry weapons with them, but don't actually use them, leaving you to do all of the heavy lifting while they provide assistance with magic. At least you never have to update their gear yourself, they get better stuff as they level up.
Marcus Fenix of Gears of War fame carried a knife on his right boot, but is never used.
Halo has an unlockable katana that can be worn on the back, as seen in the trope image. It naturally cannot be used in game. Also, the CQB armor has a knife on the front of the chest which cannot be used.
Averted in Halo: Reach, where the knife on a character's shoulder can be used for assassinations.
In Freedom Fighters, Christopher Stone gets a knife on his left shoulder, but his only CQC weapon is his wrench.
In the first Samurai Warriors game, Shingen Takeda has a katana sheathed on his belt, which he never uses. This was removed from subsequent costumes. In the second and third games, Mitsunari Ishida keeps an unused dagger sheathed hanging out the front of his coat.
Aeolus from Mega Man ZX Advent carries a sword with him all the time, but he never uses it in combat.
Although, from the transformation sequence, it appears that the sword is used as the base for his transformed model's swords, hence a reason to carry it around with him.
Adray from Star Ocean 3 equips and prominently carries around a katana, but most of his attacks are a combination of magic and punches and he never takes it out of its sheath.
Assassins from Guild Wars are extremely guilty of this. Many armor sets (including the no-armor-equipped underwear, for women) feature several daggers strapped all over their body that are only textured on, not even in model. The assassin hero Anton is even worse: he prominently wears three katanas on his back that he never uses.
Champions Online allows the player to customize their costume including some weaponry such as daggers on the belt or a sword slung across the back. However because these are costume items, they merely show up on the player avatar and don't get any particular use (not even if you have the same weapon as what you're carrying).
In Team Fortress 2, Pyro's bandolier has napalm grenades and the Soldier's belt has frag grenades. The former are always useless and the latter are only used in the Soldier's Suicide Attack taunt.
In Pocket Bomberman, our hero is seen in garb reminiscent of a Roman soldier. He has the sword and everything! He is even seen unsheathing it and holding it up heroically to the sky! But can he use it on any one of the dozens of monsters that are out to maul him? No.
In Persona 4 Arena, Akihiko has a knife strapped to his hip that he never uses. The shoulder holster is probably holding his Evoker, which isn't really a weapon.
In the Soul Series, customizable characters will sometimes get some. In 4, there is a pirate's belt that has a pistol on it, as well as gloves that have a claw on it. The only fighting is done with weapons.
In Fire Emblem Awakening, some classes are guilty of this. Assassins have extra daggers strapped to their wrists and Swordmasters are carrying two extra sheaths, yet they will never use these extra weapons even if all the weapons in their inventory break.
Strobe from Demon Fist has that huge sword on his back, but fistfights his enemies. Apparently he swore an oath or something not to use it.
NJ from Electric Wonderland always carries a katana despite having no formal training on how to fight with it.
Swords have been this for a long time and in certain situations (like a formal officers party, perhaps) to this day. In the 18th century, a well-dressed gentleman was almost required to carry a smallsword on his hip, though carrying a walking stick was an acceptable alternative.
In the Ottoman Empire it was expected that a noble be allowed to keep his personal sidearm at all times, even at formal parties. This got a bit dicey when personal weapons shifted from swords to pistols. In the end it was compromised with highly ornamental pistols, that usually couldn't shoot (although they were heavy enough to be used as a blackjack and many had a blade under the barrel).
Most weapons in the Arms and Armor gallery at the Metropolitan Museum of Art, or any other art museum that displays weaponry, were probably ornamental weapons (and some are known to have been ornamental weapons, like an Ottoman sultan's processional sword). Swords that the average person carried into battle rarely survived to be museum-quality art, even if they had been when they were made.
Several American Civil War generals never drew their swords, not even to command their men. One is said to have had his sword rust in the hilt, while another didn't even bother taking one with him, instead strapping an umbrella to his waist.
Souvenir and decoration swords. Enthusiasts speak rather of SLOs, or Sword-Like Objects. They look like swords, are ornamental and look flashy, but they are horribly badly balanced, made from inferior materials (such as stainless steel, which is prone to shatter and can't keep its edge for a long time), either over-tempered too stiff - or not tempered at all - and are too weakly built to use. They are purely wall-hangers. Even a complete layman can distinguish between a sword-like object and a real sword by merely grabbing one.