Can I be sued if I am the sole owner or a cosigner on a car involved in an accident?

In nearly every state the owner of a car can be sued for injuries caused by any person driving your car with your actual or implied permission.

If you are not any owner but merely a co-signer on a loan to the bank, that's raises a host of different issues altogether. All sorts of facts and circumstances could be marshaled to argue whether or not you should be treated as a quasi-owner and have liability attributed to you. If you were a total stranger that merely guaranteed the loan (think the equivalent of mortgage insurance) you'd not have liability.

If there is liability insurance on the car your insurance company will defend you and pay any judgment up to the dollar limits of your policy. I sure hope you opted for a large liability policy limit and/or had an umbrella policy.