What happens to costumes when tour is over?

Everybody does this differently.

A singer in a band doing their very first tour as a supporting act might either buy their own clothes, or make them themselves or rent them from a theatrical costume rental.

A major star will almost certainly have a costume designer. The star and the costume designer will meet, exchange ideas and then the costumes will be made by one of the companies specializing in this kind of work.
Michael Jackson collaborated for years with his designers:

From his white fedora to his U.S. patented "lean shoes," Michael Jackson's larger-than-life image came out of a 25-year-long collaboration between Jackson and his longtime costume designers, Michael Bush and Dennis Tompkins. "His costumes had to perform onstage as well as he did," Bush tells Rolling Stone. "They had to be like his second skin – wearable art accentuating both his style of dancing and the beat of the music."
Source: https://www.rollingstone.com/music/music-lists/the-king-of-style-dressing-michael-jackson-10590/

After the tour anything can happen to the costumes. They might be thrown away or go into storage somewhere. The artist or their management might keep something as a souvenir. Sometimes these will be auctioned off years later. Two of Madonna's iconic cone-shaped bras from her 1990 "Blonde Ambition" tour were sold at Christie's in London for $77,000 in 2012. Sometimes they'll end up in a museum: Michael Jackson's outfits from his 1984 "Victory" tour were part of the inaugural exhibition at the Smithsonian "African American History Museum"