I suppose the subject says it all.
Though I don’t know what the general practice is today, when I got my security clearance in the 1970s, it was considered the right of every citizen by law (though they didn’t exactly advertise that fact). Today, clerks and other less well-paid employees often get high security clearances at their employer’s expense, but better paid employees are often required to pay up to a few thousand for their investigations.
In the 1980’s, we moved the highest security items to the SCI [Sensitive Compartmented Information] system, which I believe is stilll in place today. This was essentially a ‘need to know’ system, based on SCI codewords. Instead of being cleared to a certain level, you were cleared for certain projects to a certain level.
I would imagine that most First Ladies would qualify for a high security clearance (though it’s interesting to wonder what would happen if they were a patent security risk) and which would be investigated and granted as a routine part of their husband’s inaugural preparations. (IMHO security clearance investigations range from uselessly routine to ridiculously paranoid, with a mean value of simply ‘ridiculous’) Considering that Presidential candidates are not required to meet any security prequalifications (and I happen to think that’s a good thing overall), I’d guess that no one would bee too eager to deny a First Lady a respectable clearance.
On the other hand, the First Lady would not automatically be cleared for any specific SCI information. In fact (based on what I heard, back in the day - a lot of which was probably garbage), the President himself is not automatically guaranteed SCI access to all SCI secrets [i.e. he can’t just decree "for I am the President, bring me all the finest secrets in all the land!] True, I wouldn’t want to be the Classifying authority who told the President -or any Cabinet or White House official- s/he wasn’t cleared, but it’s still technically a possibility, and dark rumors abounded (Formally, each Classifying Authority retained official control over who accessed its secrets)
So the First Lady might not get access to atomic bomb plans, but as a practical matter, I’d say nonspecific pillow talk would be legally safe. The security training I got (they don’t just give you the clearance because you’re so delightful; you have to sit through annoying security films not unlike the stuff you watched in school, and more than a few lectures) did seem obsessed with pillow talk