Library of Congress' Music Library
Many of the songs that F.S.F. refers to in the novel are available on the Library of Congress' web page. They have public domain, original recordings of the songs mentioned. For example, here is their copy of the song "Three O'clock in the Morning," which is playing as Gatsby leaves his party in chapter six:
http://www.loc.gov/jukebox/recordings/detail/id/8979/