I may add more later.  However, a good choice if you are under age 26 is to pick pick up the form from the Post Office and register wilth the Selective Service System.  If you pick up the form from the Post Office, you are not required to provide a SSN.  It is a very good secondary form of identification. 

You should use a mail drop and for extreme safety, I suggesting getting someone else to fill it in and sign it.  Pay a child or transit $10.  The risk is extremely low because that is not what people do.  Therefore, it is very unlikely that any government clerk will question it.  In about six weeks you will receive the draft card in the mail.