I read online to wait 2 weeks after landing to apply for an SSN due to the system needing to update with the documents I had on me at POE (K1 visa.) I got married 5 days after I landed, and applied for my SSN after the 2 weeks with absolutely no issue. I made sure to take all of the correct information including my marriage certificate as I changed my last name to match my husband's so it was different than my passport and the process was seamless for me getting an SSN, I got mine in the post about a week later. I then was able to open a bank account easily. I was told by the person at the SSN office once I've got my green card to go back and change my SSN card from one where I can't work, to one that can.
Not sure if it's different for each office but I had no issue at all!