Though we're now given Social Security Numbers at birth, many people had to register for a number when the Social Security Board began giving them out in November 1936. Interestingly enough, the Social Security Administration states on their website that the first "Social Security Record" was created for a John David Sweeney Jr. of New York State. (Source: History of the First SSN Card)
Applicants had to fill in the SS-5 form with the following data:
2) Maiden Name
3) Current Address
4) Birth date
5) Birth place
6) Father's full name
7) Mother's full name, including her maiden name
In my case, I had a brick wall in my great-grandmother, Irene (Stephens) Rodgers who was born in 1902 before Missouri's statewide birth certificate registration really took hold, and of course, there was more than one Irene Stephens/Stevens of similar age in Missouri during the 1910 census.
I knew her date of death (1973) and where she was buried, but I hesitated in spending money on ordering a death certificate for one very good reason. The thing about death certificates is that the information on the deceased's parents is only so good as the informant giving the information.
Enter the Social Security Death Index (SSDI). I had trouble finding Irene Rodgers in the database, and then I found a woman with a different surname but correct death date. That was when I found out about her other two marriages that I never knew about. Armed with her social security number, I decided that I would order her Social Security Application. It cost $27.
A couple of weeks later, I received a copy of the form in the mail. The application provided me with the information I needed, and as a result, I found her parents' death certificates, photos of her parents (my great-great grandparents) on findagrave, and extended my family tree back a few more branches. I would definitely recommend giving this a try.
Social Security Death Index (SSDI) (Via Family Search Record Search):
Social Security Application Ordering Form (Via Social Security Administration)