Apparently some people still carry their Social Security Card. I used to do that but stopped about 25 years ago. These days, it’s foolish at best to carry your card. It’s too valuable. If you lose your wallet and your card is in it, you have a long-term big problem. At a minimum you have to get a new SSN and notify everyone. This can be a major pain. It could also lead to major confusion. So, don’t carry it.
Now some employers in these days of immigration hysteria may make you produce the actual card itself. If so, use it and then put it back in safe-keeping. Don’t carry it!
BTW: I blogged over a year ago about what to do if your identity was stolen. You might want to review those posts if your have that problem.