This is an entirely hypothetical question, of course. Assume you’re going through the papers of a deceased relative and among them are documents that seem to be marked Top Secret or titles of that kind, and after some Googling they seem to be accurate.
What should you do? Call the FBI?
If the government’s on its game, FBI agents should already be surveilling your home.
Hypothetically speaking, of course. 
*yes, contacting the F.B.I. is what a responsible citizen would do, not go on social media to quote selected passages and brag on your access to secret information, sell the papers on eBay or store them under the hot tub. If I was hypothetically in that spot and wished to avoid hassle, I might be tempted to hypothetically burn them.
Roll it up and smoke it.
Kidding, ha ha. I would just call the cops in my town, and let them ratchet it up the food chain as they saw fit.
(This is after I had contacted my attorney.)
Scan it real quick for mention of Roswell or the JFK assassination…then call the FBI.
I would probably try contacting the agency from whence they came. I would assume they would have some form of marking indicating that.
No way. That would mean you looked at something you weren’t supposed to. (only semi-kidding) If they’re clearly marked Top Secret or such, FBI or the local Congresscritter’s office.
You think the FBI wouldn’t think that?
Staple it shut, since the cover page probably doesn’t have a lot of confidential information on it, then return it to the place it was from. If they want to call the FBI, they can.
Does a random citizen have a duty not to read it? I know I can’t refuse to return it , etc. but I don’t think reading it is a crime. I could be wrong, of course. I haven’t researched the question
I look at it differently from others.
In order to do the googling, it sounds like you read them.
If you have good reason to believe that the New York Times or Washington Post would want to print some of the secrets, and you yourself believe it is in the public interest, then you should contact one of those papers. I believe those papers will be careful not to print parts of the documents the disclosure of which have high risk of endangering lives.
If I strongly doubted that disclosure would interest our best newspapers, or was in the public interest, I would call the lawyer referral service of my county bar association and see if they could suggest an attorney I could discuss this with. Ask your referral service about this, but the first legal appointment should be be low enough cost that almost all Americans can afford it.
Then I would follow my attorney’s advice so long as the price of any needed retainer was affordable to me. Even if all I could afford was the initial consultation, I think I would be better off than going to the government with zero advice of counsel.
I gotta admit that if I found some top secret papers among a deceased relatives things, I’d take a quick gander at them before turning them over. If I didn’t, the curiously over what they were and how my relative got a hold of them why they kept them would eat me for the rest of my life.
Most of the TS documents I’ve handled were boring – it was the Secret stuff that was interesting. 
I live near a Navy base; I’d call the local NCIS office to come get it.
If it is highly damaging to the government and you read it, you become a problem to be solved.
If it is highly damaging and you didn’t read it, they would have to assume that you might be lying about it…and you become a problem to be solved. It is a matter of legality vs. self-preservation.
This.
I have a secret clearance and SIPR account. It’s boring shit to 99.99% of people. Just meaningless numbers and stuff. Move along people, there’s nothing to see here.
So would I. If I ended up spending the rest of my life in an unreported detention cell, I’d at least want to know why.
And if it is interesting information to a foreign entity and they learned that you read it, you are a resource to be exploited. Safest to burn before reading.
The thread question is about what you “should” do. And I don’t generally think you should appease a foreign entity.
On the other hand, we are in factual questions, and “should”, as I interpret the word, is an opinion, not a fact.
If you should act for maximum safety, the U.S. government is more of a safety threat then a foreign one. So the safe thing is to consult a lawyer who practices in federal courts. Ideally the lawyer knows something about national security law. If you can’t afford that, or there are none in your city, get some kind of legal advice. Then if they say you did wrong, your defense is that it was all on advice of counsel.
I don’t think you understand what I said. What I meant is that if you read a secret document with useful info and admitted that you had possession of said document to the authorities, there is a possibility of somebody deciding that the way to gain access to that info is to grab you and torture it out of you. Therefore it is better that nobody ever knows that you saw it.
I’m in Canada so your jurisdiction may vary but I once worked as a sub-contractor for the Federal Government processing data related to commercial fishing. This data was not any kind of military secret or national security type of stuff and would not mean much of anything even to people who would know how to interpret it. But where there were cover pages they were explicit about the various kinds of legal proctology one would be in for if they proceeded past this page. Now how that would actually be enforced if a person was to hypothetically find a stack of these documents somewhere is not entirely clear to me. But if anyone was to ask you if you had looked, say no and mean it with the entirety of your being.
If you are not a spy or a government employee who has mishandled classified documents, I would not worry too much about top secret documents, especially if they appear on Google, which means somebody already leaked them to the public. In that case, it is not clear from the description whether your relative was the source of the leak or just printed them out from the New York Times’s web site.
Of course I’m sure I would read it. Afterward, I would destroy it and never speak of it again and get on with my life. I prefer to avoid dealing with the burdens of bureaucracy because in this case, it involves too much of unknown consequences.