By law now in the U.S. you have to show the SS card when filling out the I9 for employment. Many companies now will verify that you are legally eligible for employment prior to hiring you.
That isn't entirely true, those filling out I-9 paperwork must present either 1 form of ID from column A (like a passport) or 2 pieces of ID-one from column B and one from colum C (like a drivers license and social security card). According to 1990 law employers cannot require to to present additional or other forms of ID for I-9 purposes. Employers can, however, ask to verify your social securtity card for tax reporting purposes, but this is unrelated to I-9.