The Master Sword in its Scabbard
from Twilight Princess HD
A Scabbard is a type of sheath used to hold swords. In The Legend of Zelda series, especially in the console games, Link usually has a scabbard for his swords. The most ornate scabbards tend to be for the most advanced sword, which is usually the Master Sword.
Link infamously wears his scabbards clipped onto a shoulder belt instead of at the hip. This, along with his choice of keeping his shield on his back, creates his signature image that is instantly recognized. For most swords, the scabbards are automatically changed with them. Though a select few share the same scabbard. Some scabbards have mouthes, the name for the opening of the scabbard, that matches guard. These parts of the sword form what's known as a chappe, which with at fitted mouth makes a seal when the sword is in the scabbard. The swords and scabbards that have these attributes tend to be the most ornate.
Ocarina of Time
Ocarina of Time introduced scabbards to the series. The scabbard changed depending on Link's age, so the Kokiri Sword had one scabbard and the Master Sword, Giant's Knife and Biggoron's Sword shared a different one. In Ocarina of Time 3D, the latter two blades were given a unique scabbard to match their longer appearance, and a strap was added to Link's model as shown in official artwork.
Majora's Mask had a scabbard for all the normal b-button swords. The Great Fairy's Sword and Fierce Deity Sword lacked scabbards. These scabbards are notable for having very accurate sizes in regards to the sword they hold. The Gilded Sword's scabbard is absolutely massive and sometimes clips the ground as Link moves. It's also the first in-game appearance of Link's shoulder-strap.
The Wind Waker
The Wind Waker had two scabbards, one for each of the swords. All the stages of the Master Sword used the same scabbard. These scabbards followed the style of The Wind Waker of being cel shaded smaller in size.
Twilight Princess had scabbards for two of the three swords. The Wooden Sword was only held by straps instead of a full scabbard. These scabbards have more detail in them than any other game before or since.
Phantom Hourglass is the first handheld game to have scabbards. Both of the swords have small scabbards, though Link's hat consistently obscures most of them. Link's initial scabbard given to him by Oshus is identical to the scabbard seen in The Wind Waker.
Spirit Tracks has scabbards for both swords. Though again Link's hat consistently obscures most of each scabbard.
Link's Awakening, Four Swords, Four Swords Adventures, and The Minish Cap do not depict Link with a scabbard in-game, usually because his sword is always drawn. However, Link is depicted with a scabbard in artwork for the games. His scabbard in Link's Awakening appears as a dark brown with a red jewel at the bottom. The scabbards seen in Four Swords, Four Swords Adventures and The Minish Cap appear identical to the Hero's Sword's scabbard seen in The Wind Waker.
Sometimes other characters carry scabbards for their swords. One notable scabbard here is the one for the smaller sword of a Darknut in Twilight Princess. When the Darknut pulls out this sword, its scabbard gets thrown to the side. There is also the scabbard that Ganondorf uses to carry the Sword of the Sages.
Ashei carries this sword in its scabbard at all times, and is never seen taking the sword out of it.