Normally, you sell the right to your songs - such as a publishing deal where the publisher has the right to the song for x (let's say 24 months). At the end of that time, all rights revert back to you. You don't actually give up the copyright - you simply give permission for other to use the song and you agree to share some percent of income (publishers nomrally get 50% of any income)
As indicated you can trasfer a copyright (for a fee) - which is in essense, selling your song.
In the old days (back in the 60's & 70's) it was not uncommon to sell songs - often people would play a song for someone, the listener would like it and offer $50 - $100 for the song. However, that type of arrangement is rare.
While I agree it may be a good idea to "lawyer up" - the fact is that costs money, often more than a poor writer has. My personal opinion is, unless you are absolutely possitive you have an amazing songs and are possitive you have the connections to get it into the right hands - it may make more sense to make something (even if you give away some rights) rather than make nothing (and have some bragging rights to a song that no one will even hear/publish/record).