It's interesting to see the views expressed , and naturally, slightly conflicting views which probably is attributable to two groups being on different sides of the Atlantic.
Some of you appear to be speaking with a degree of legal knowledge but I would challenge M.Teds assertion that "You have Certain fair use rights related to any song, whether it is in copyright or not".
Of course you would have rights to a song that's not copyrighted - (which I can't envisage), but I believe you wouldn't have any rights to a song , product ( or a face as Rara Avis explained ) or indeed any other thing which is copyrighted.
Breach of copyright is not simply a Civil matter. To play music (in any form i.e. records /disc /live etc) in public, let's say a bar, if you have not paid the PRS licensing Fee for the establishment - the PRS can inform the Police who will act swiftly to ensure the Fee is paid , under pain of stopping the music altogether and possibility closing the premises. Payment is usually made very quickly.
A guest informs us that "in the US if a song has been recorded it is considered "published"" - I believe over here that a song which is sang in public, registered with the PRS and by providing them with the Lyrics and Score it is properly protected - it does not depend on whether is it a recorded work, or not, and lets face it many of us have written good songs which have not been recorded, but which we intend to retain full rights of Ownership.
I can only assume that it must be most daunting for our American friends, going around clubs, singing their own songs and not knowing if they will be stolen before they can establish ownership by involving a Publisher.
I'm enjoying reading you r views – but the reoccurring theme /advice is – contact the owners of the song (s) - who knows ?? - you may even make new friend(s) through doing so.