My advice:  After you get married, hand carry the completed marriage license to your County Clerk or whoever registers the marriage.  Get a registered marriage license.  Take the license (along with the other paperwork required) to your local Social Security office and get the SS card.

And do this before your fiancee's visa expires as the SS website specifically states that you must have a valid visa.

By getting the card after marriage, then the card is in your wife's married name.  If you get it before marriage, then you have another trip or more paperwork to fill out at the SS office to change the name.

The reason I am posting this advice is that I have seen a number of people on the this list and other lists that have waited until the visa has expired and they have had real problems getting the SSN.  After visa expiration, the SS office requires some other paperwork associated with your AOS and that may be hard to obtain.

IMHO