A) A U.S.Citizen
B) An alien lawfully admitted to the U.S. for permanent residence or under other authority
C) An alien who is legally in the U.S. but not under authority to engage in employment (i.e. non work purposes)
A non-work purpose can arise where the need for a SSN to satisfy a federal or state statute or regulation that requires a SSN to be eligible for a federal or state benefit.
An applicant for a Social Security card must submit documentary evidence. An in-person interview is required of any applicant 12 years or older applying for an original SS card. There is an exception for an alien who requests a SS card for immigration purposes.
You can actually get a new Social Security Number in certain situations. If you have deep felt religious beliefs and the numbers in your SSN are in conflict or are offensive (i.e. 666) then your number can be changed. Also, victims of domestic abuse where harassment/abuse or life endangerment is alleged can get a new SSN. There is a requirement that written affirmation is obtained from a third party (police, abuse shelter, etc) to confirm the domestic abuse.
The Agency can now charge for replacement of lost or stolen Social Security Cards. Don't leave your Social Security Card in your glove box. Keep it somewhere safe and out of sight.