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Several things can happen to the clothes. In reality they aren't considered "clothing" such as you buy at the store. They are technically costumes, even if it looks like what you find at walmart or American Eagle. After production wraps costumes are stored in case they need re-takes or re-shooting. After the release many times costumes end up in storage. There are large warehouses filled with costumes and clothes used in movies. Important pieces are held for display and publicity, but most end up in storage. Sometimes costume departments for other movies will recycle old costumes, re-making them for a new character. When the warehouses get too full the costumes will be culled, some sent to thrift stores, some will be auctioned off, some will be sent to film museums depending on the importance of the costume and the actor who wore it. Some actors have riders in their contracts that allow them to keep a few special costumes, usually ownership belongs to the production company. Costumes are considered an asset and something that can be sold to raise money for other productions.

For television programs all costumes are kept until the show wraps up production, then they are dispersed. When productions shuts down the actors are given first dibs at their favourite props and costumes. Julia Louis Dreyfuss kept her entire Elaine wardrobe from Seinfeld; Jason Alexander kept George's glasses, sneakers, velvet clothes and the Yankees sheets and blankets. The girls from Friends kept many items and auctioned off many others as part of a fundraiser. Important pieces are held for publicity and museums (I imagine Horatio Caine's sunglasses will end up in the Smithsonian, like Seinfeld's puffy shirt) and the rest are auctioned off (important pieces again, think Sex and The City) and the rest sent to thrift stores.

Stage costumes are not like real clothes in the materials used and the construction. A 18th century court dress may look authentic on stage, but it can be constructed on modern fabrics with invisible zippers to allow the actor to change quickly between scenes. Stage costumes are they are kept in storage and/or recycled into new costumes. The Stratford Festival in Canada re-uses costumes, laces, trims, shoes, wigs etc. There are a few important costumes that are kept for publicity. Every few years the costume department hold a garage sale of costumes, laces, shoes, and grab bags of leftover supplies. I have been to several of these sales and know people who work in the costume shop. They make everything from dresses to wigs to shoes.

It really depends on the costumer and the production company. There are cases where costumes are merely trashed, but they have come to realize that people are willing to spend lots of money on old costumes.