Does the department of homeland security have rules that says an immigrant with permanent residence?

can not sign a signature card of his or her husband's or wife's checking account based on not having a social security card?

You can sign it, does not mean it will be used, never know with your bank. But getting a social security card would be good pay taxes and get benifits in the end, when you retire.

If they were granted permanent residency, wouldn't they have then been issued with a Social Security Number? I'd follow up on the paperwork and find out firstly if they were entitled to one, and what to do about getting issued one, to avoid such problems in the future.

can you have a joint checking??? I think this is your question. If you don't have a SS# I think you can still bank here in the US and have a joint checking account. But also if you are a perm resident you CAN apply for a SS#

haven't heard of that, i dont know if it differs for different states. I am a permanent resident and I do sign the signature card, but if you dont have your ssn that might be the problem and not because you are a resident.