Stays

In regard to clothing, "stays" provide structure or stabilization to the fabric.  In regard to dress shirts, the stay helps the collar "stay" put.  Meaning stay in place and flat.

Collar stays help provide structure or support to the collar of shirts.  A good example would be the use in some types of dress shirt.  Note: generally not used in the button down color dress shirts.

• A collar stay, collar stick, collar tab, collar stiffener, or collar stiff is a shirt accessory consisting of a smooth strip of rigid material, rounded at one end and pointed at the other, inserted into specially made pockets on the underside of a shirt collar to stabilize the collar's points. The stays ensure that the collar lies flat against the collarbone, looking crisp and remaining in the correct place.
• Collar stays can be made from a variety of materials, including metal (such as brass, stainless steel, or sterling silver), horn, baleen, mother of pearl, or plastic. Shirts often come with plastic stays that may eventually need to be replaced if they bend; metal replacements are less likely to have this problem.
• Collar stays are removed from shirts before dry cleaning or pressing.

Learn more about shirt collars.

In corsetry, a bone works similar to a stay in the fact that it provides support or stabilization to the garment. 

• The bone is one of the rigid parts of a corset that forms its frame and gives it rigidity. The purpose of the boning in a corset varies slightly from era to era. Generally, the cinching/shaping properties of corsetry puts strain onto the fabric from which the corset is made. The boning supports the desired shape and prevents wrinkling of the corset fabric. Bones, and the substances used for the purpose, are generically called "boning"; however, the name likely arises from the use of whalebone in early corsets. Today, many corsets use nylon or Rigilene boning, although steel is still favoured for high-quality corsets. Plastic bones do not have the strength required for tightlacing and are known to warp and bend, often in unflattering ways. Many modern bodices, strapless gowns and lingerie use a variety of plastic boning because it is much cheaper. Modern steel bones come in two basic varieties: "flat" steel boning (sometimes called just "enamel" or "rigid" boning) and "spiral" steel boning. Spiral boning is flattish but thicker than flat boning because of the tips required on the ends. Flat boning bends in only one direction, while spiral steel boning bends easily in two directions. Spiral steel boning may thus be used on curved channels or where a more gentle support is acceptable.

The Small Strip That Keeps a Collar Crisp

The most familiar stay in modern dress is hidden inside a shirt collar. A collar stay - also called a collar stick, collar tab, collar stiffener or collar stiff - is a shirt accessory consisting of a smooth strip of rigid material, rounded at one end and pointed at the other, inserted into specially made pockets on the underside of a shirt collar to stabilize the collar's points. The stays ensure that the collar lies flat against the collarbone, looking crisp and remaining in the correct place. Without them, collar points curl, lift and drift through the day, and a dress shirt loses the clean frame it gives the face and tie. With them, the collar stays put - the everyday meaning behind the name. Collar stays are generally not used in button-down collar dress shirts, whose points are buttoned to the shirt front and held by that fastening instead.

Material decides how long that crispness lasts. Collar stays can be made from metal such as brass, stainless steel or sterling silver, and also from horn, baleen, mother of pearl or plastic. Shirts often come with plastic stays that may eventually need to be replaced if they bend; metal replacements are less likely to have this problem, because a bent stay holds the collar in exactly the wrong shape. Care follows from construction: collar stays are removed from shirts before dry cleaning or pressing, protecting both the stay and the collar pocket from heat, pressure and loss in the cleaning process. It is a small ritual, but it marks the stay as a separate, reusable component - structure supplied to the garment, not part of its cloth.

Boning: The Stay at the Scale of a Corset

In corsetry, a bone works similar to a stay in that it provides support or stabilization to the garment. The bone is one of the rigid parts of a corset that forms its frame and gives it rigidity, and bones and the substances used for the purpose are generically called boning - a name that likely arises from the use of whalebone in early corsets. The purpose of the boning varies slightly from era to era, but its core jobs are constant. Generally, the cinching and shaping properties of corsetry put strain onto the fabric from which the corset is made. The boning supports the desired shape and prevents wrinkling of the corset fabric, standing as a skeleton that cloth alone could not maintain under tension. Many modern bodices, strapless gowns and lingerie use a variety of plastic boning because it is much cheaper, while steel is still favored for high-quality corsets.

Steel boning itself comes in two basic varieties with different behavior. Flat steel boning, sometimes called just enamel or rigid boning, bends in only one direction. Spiral steel boning is flattish but thicker than flat boning because of the tips required on the ends, and it bends easily in two directions, so spiral boning may be used on curved channels or where a more gentle support is acceptable. Today many corsets use nylon or Rigilene boning, although plastic bones do not have the strength required for tightlacing and are known to warp and bend, often in unflattering ways. From a two-inch collar strip to a full corset frame, the principle never changes: where fabric must hold a shape against movement and strain, a rigid stay is slipped, sewn or channeled in to hold it.