So if you cover a song
then put it up on Myspace.

Will there be copyright issues?

Or does performing it yourself make it your own?

Let me know
You would have your account removed. Copyright law states you would own copyright to that recording. That is, to that collection of sound waves. You do NOT however have any right to the song itself.

If the composer has been dead for over 70 years then the song is in the free domain. And anyone can record or perform it without a problem.
So what if we add to it or take things out?

Myspace is going to be gay and delete the account?
All the big artists out there now have covered songs, especially when they were first starting out. I doubt it would be a big deal.
Good question, 'Cause I have uploaded a recording, and it's my recorded original stuff, till the end, in which I cover part of a song (the end of the intro and the verse, plus part of the first solo).