Most of these sites offer free online listening with no registration necessary. They may also offer memberships (some free, some paid) where you can use a wider range of services, download, avoid ads, and so on.
Accuradio hear your choice of many varieties of music free, but with ads.
Bandcamp hear new indie music free, with opportunity to purchase from artist.
Classical Music Online hear classical music. Search by genre, country, concert, or for orchestras, ensembles, quartets, etc.
ClassicalConnect Search by instrument, performer, composer. Hear music, artist interviews, read liner notes, more.
Folk Alley Hear traditional and contemporary folk, Americana and roots music from the Folk Alley radio show.
Jamendo hear, download and share many sorts of indie music for free, at this site that works with Creative Commons.
Last.fm is a music discovery service that suggests new artists based on the music you listen to. Assorted genres, but best for pop, rock, and related mainstream genres.
Live Music Archive
Concerts, festivals, bar songs, the occasional symphony, and more - from this year and the past, famous, infamous and unknown, all preserved at Archive.org.
Live 365 Listen free, but with ads. Online radio searchable by wide-ranging genres and sub-genres (example, within Folk, select from Contemporary Folk, New Acoustic, World Folk, Alternative Folk, Folk Rock, and Traditional Folk.)
Mixcloud Hear online music radio shows, mixes, or talk radio, follow your favorite disc jockeys, search by popular genres.
Pandora music discovery with the Music Genome Project. Find songs with interesting musical similarities to your current favorites. There are ads.
SoundCloud hear fresh music and spoken word podcasts. Science, storytelling, comedy, and more. Scroll through your choices by using the "category" box in the top right corner of the page. SoundCloud is a social platform where anyone can post and share.
Spotify sign up to hear music free, including many of the biggest, most popular artists and genres. Search by artist, album, genre, playlist, or label. Expect ads.