Social security number is a unique identifier for a person authorized to work. As soon as an individual lands in US with H1 visa it is his company responsibility to apply for Social Security Number. Spouse and dependants are not eligible for SS number though in some states SSN is assigned with - Not authorized to work - stamped on it. For opening a bank account SSN is not mandatory and SS Agency issues a letter in this regard. SSN is must for applying for Credit Card also as the card issuer(s) check the credit history of the applicant using SSN.
Documents required for SSN application:
b) Work Permit
Once the Green card is issued SSN can be obtained. For it the documents required are:
b) Green card
Please follow the link for more or to locate Social Security Agency near your place