One method for accessing free '50s and '60s music online is via radio websites, such as Live365.com and AccuRadio.com. Another way is to access oldies music lists on YouTube.
Live365.com is an online radio network allowing broadcasters to distribute their radio stations online. The website features dozens of radio stations related to '50s and '60s music as of 2015. These include stations such as 57 Chevy Radio, which is dedicated to hits from the late '50s and the '60s, as well as Sock Hop Oldies, a station playing both classic hits and rare songs from those two decades.
AccuRadio.com is another online radio network with oldies stations. The site allows visitors to listen as guests, but it's also possible to register for free. Registering allows members to listen to music with fewer commercial interruptions and customize channels to play the music they prefer.
8Tracks.com is a free online platform for users to create playlists. The users dictate the content of the playlists, and several are related to '50s and '60s music.
A similar service comes from YouTube. Users upload songs or create playlists out of other people's uploads. YouTube is searchable by keywords, and visitors can access lists related to '50s and '60s music. It's possible to listen to the whole list created by another user.