Ornamental Weapon

Do you think that sword's permanently glued to his back?

"That thing on your back isn't ornamental, is it?"

—Rod (to Rue), Threads of Fate

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.

Examples of Ornamental Weapon include:

Anime and Manga

Kouga in Inuyasha carried a katana for ornamentation.
- 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 The Familiar of Zero 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.

Literature

The Big Bad in the Left Behind series of books carries one of these during the war of Armageddon. The sword is constructed to appear Too Awesome to Use, so it just gets waved around a lot. The books make it clear that when the Big Bad really needs to kill somebody, he uses a gun.
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.

Video Games

Angeal Hewley (pictured) from Crisis Core -Final Fantasy VII- 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…
- The Buster Sword did end up getting rusted… when Cloud used it as Zack's gravestone.
  - 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).
Ganondorf's sword in Super Smash Bros.
- 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.
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.
Marcus Fenix of Gears of War fame carried a knife on his right boot, but is never used.
In Metal Gear Solid 3: Snake Eater, our protagonist runs into a young man going by the name of Ocelot. On their first meeting, he's using an automatic pistol, but using revolver techniques. After getting scolded for that, he returns later with an engraved revolver, which is ridiculously classy. He's again scolded, this time for bringing a pretty yet impractical weapon into combat, being told it would make a decent collector's item. Thereafter, he fights with his now trademark plain Colt Single Action Armies.
In Tekken 6, there is a character customization feature that lets you equip your characters with sheathed weapons, only for looks or stats in campaign mode.
- Most Fighting Games that focus strictly on hand-to-hand combat do this, usually when there's ninjas.
Speaking of which, ninja Kagemaru in Virtua Fighter can have a sword. Drunken Master Shun Di can have a hermit staff as well.
Halo has an unlockable katana that can be worn on the back. It naturally cannot be used in game.
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.
Aeolus from Mega Man ZX Advent carries a sword with him all the time, but he never uses it in combat.
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.
The online superhero MMORPG, 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.

Real Life

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.
Renaissance faires and fan conventions, given that weapons (both obviously fake and looking real) are an inevitable part of cosplay. Most adopt variations of "peace bonding" - ways to visibly ensure that whatever it is, no one can draw it. The term apparently comes from Canadian law. Sometimes also used for mark-of-approval tags on props.