The good news is that Facebook now allows sharing MP3 files on profile pages. That means, if you want to play a song on your friends profile page, just paste the URL of an MP3 file which is publicly accessible.
There are third party apps like Muziic, which makes it easy to share songs with your Facebook friends. There is also Divshare, which lets you share files with Facebook buddies.
But the problem arises when you want to share a song from your computer because Facebook still does not allow users to attach or upload MP3 files in friends wall or timelines.
The quick workaround is: upload the song to any file sharing services like Dropbox. Just make sure the link to the file is publicly accessible and the song ends with an MP3 extension. Next, copy the URL of the file and paste it in your friends wall. Your friend or anyone can then play and listen to that song from their Facebook profile. You can also use this trick to record audio online and share it on Twitter or Facebook.