I've noticed a lot of hits on Alien Spouse coming from people doing a Google search on "I've Got My Green Card But I'm Still Waiting for My Social Security Number", or variations on that theme.
I was in that very position for months, and I sympathise.
Essentially the problem is that there has recently been an unprecedented and massive spike in immigration and citizenship requests. This is having a knock on effect on the time it takes to issue Social Security Numbers to immigrants, because someone thought it would be a good idea to tie that additional piece of red tape into the immigration process.
Basically the attempt to streamline the system has backfired somewhat, and it's entirely possible to be waiting for months for your SSN.
My advice is two fold.
First if you are calling the main SSN helpline make sure you have someone to hand who already has their own Social Security Number. The automated system will not allow you to speak to an actual person without that number. Once you are through to a real live person in a real live call centre, they will only be able to tell you if your Social Security Number has already been processed and is on it's way, nothing else.
Secondly use this webpage from the Social Security website to locate contact details for your nearest Social Security office. Call that office, and specifically ask if you should come in person to see them and perhaps fill out another SSN application form. They will not suggest this course of action themselves, you must ask if that is the best thing to do in your particular case.
In my case going to the local Social Security Office got me my SSN in exactly seven days, but obviously times will vary according to the individual circumstance.
I've written about my experiences of trying to get a Social Security Number a couple of times, you can find those posts in the Green Card and Visa Category (which this post is now part of), and also on the page How I Became an Alien Spouse.
I don't know how much practical help any of what I've written will actually be to anyone, but I hope it might make you feel a bit better about it all.