It is much more complicated and risky than you would imagine. Of course, some fraudulant card device or visual imaging that copies the card information is the most obvious attack. Then there is corporate hacking that breaks in the store's credit files in a massive manner. Then there are the unknowns of the internet world (with spy ware, etc.). In sum, it is a very dangerous world out there. Trust me, it is very easy to get one's social security number and address via various means after one gets hold of the credit card information. Cyber attack is the most present and worst danger for us all.
I fully agree with Lou on protecting oursleves from ID through all means. Further, I just subscribed the Amica's ID Protection program with daily monitoring and ID theft resolution support and insurance coverage today (in addition the Creidt Karma). I may subscribe to more comprehensive Identity Guard or LifeLock (with monthly fees) after I retire.
It is the unseen risks which are most fearful and damanging.
Just my input.
Here is an informative article: http://money.cnn.com/2008/08/26/pf/identity_theft_prevention/index.htm