Applying for a number: You apply for a Social Security Number by filing an SS-5 form with the Social Security Administration. The Social Security Administration's posts SS-5 forms .
Finding Social Security Numbers: If you have a person's name, you might be able to look up his or her social security numbers using Accurint, TLO, KnowX and/or LocatePlus. However, you should probably check with the company first, because the number will probably be masked for privacy purposes. The database vendors may be willing to unmask the data for certain businesses (e.g. collections firm).
If the person might have died, search the Social Security Death Index on Ancestry.com.
Other places to find social security numbers: court records, death records and professional licensing records. You might also be able to get a social security number by hiring a private investigator.
Validating a Number: If you have a social security number, the SSN Validator will tell you whether that number was actually issued and, if it was, where and approximately when, and possibly whether there is a related death record. You could also register and use the government's official Social Security Number Verification Service, or you could run the number through the databases discussed in the entry for "Finding People."