Handcut Designs has a page referencing 21 Free Music Players For Your Website.
There are no reviews on any of them. I tried and stayed with the first one: dewplayer. It is easy to setup. I used the version which presents a playlist. Currently, I am using it in a private playlist. Once the band (Radios in Cars) releases their music, I'll provide a public playlist.
As a point of reference, playlist files are described in a specification provided at XSPF.