In my past experience, USCIS likes to disregard the marriage certificate as having any legal meaning until they approve the I-485, so they tend to keep using the premarried name until the time comes to issue the green card.
You can get an SSN using the K-1 visa only during the 90 day period of authorized stay hat was granted upon admission. That's only because the K-1 visa automatically authorizes employment incident to status. They do this because many states ask for SSN on the marriage license application.
If you're still within the 90 days, or you've received the work permit, you can apply for a SSN. However, I would not follow the earlier advice about saying you're still single (USCIS may later see that and deny your I-485 as a fraudulent marriage), rather I would indicate you are married but want the SSN issued or verified in your primarried name. SSA should be able to do that because a marriage certificate usually only authorizes, but does not require, you to change your name to your married name (i.e. you don't have to use it, but once you do, you can't change it back).
You should get a SSN as soon as possible. You will probably need it for a variety of things, such health insurance, bank accounts, credit, taxes, employment, etc.