Usually, once you apply for the SSN they really are just waiting for the physical card to be sent to you - as they do not print these at local SS offices.
I think most people had issues at the SS office after getting married because local clerks think that by getting married you 'VOID' your visa...when in reality the visa is valid for 90 days from entry (somewhat regardless of date of marriage). I think most people had issues when the non-USC spouse was the one changing their name upon marriage, as it raises a few eyebrows.
We are planning to get my fiance's SSN after marriage, as we really have no choice with him flying in so close to our planned wedding date, as well as the holidays. He won't be changing his name so hopefully there are no issues.
This is all based on things I have read on here though! Might be different experiences by state.