I was browsing someone's thesis from a US university. I found several typos and missing (incomplete) sections. There was no plagiarism and dishonesty though. What would happen if I write to their advisor? Can some action be taken or is it a closed case once a thesis is accepted and signed.
Generally no official action can or should be taken regarding errors in a dissertation (assuming they are not a sign of dishonesty or fraud). However, it could be worth pointing out errors to the author in case he/she is preparing a publication based on the dissertation. I wouldn't do this if it's from long ago or you see that the material is already published, but you could be providing a useful service to the author otherwise. (If these errors have already made it into publications, then it's worth reporting them to the authors if they are substantive, so they can decide whether to publish errata, but it may not be worth reporting typos. I'm a perfectionist and would like to know of typos in my publications, but others might be annoyed.)
But you certainly shouldn't write to the advisor. That comes across like you are reporting bad behavior to an authority figure, so it's not appropriate unless that's the message you want to send (for example, if you discover plagiarism). Instead, you should communicate directly with the student who wrote the dissertation. If you can't figure out how to track down the author, then you can assume he/she isn't pursuing a research career and you don't need to worry about it.
I do not know really. After all, it is the adviser, who bears the primary responsibility for checking the results of the thesis (the student is not mature enough most of the time and the rest of the committee do not care much this way of the other).
My personal experience with this was that my adviser gave me a 50 page thesis of his former student (written like 5 years before) and asked me to close the gap between the lower bound of $m$ and the upper bound of $m\log 1/m$ for some quantity depending on the small parameter $m$. With my usual laziness, I decided just to see what I can get myself instead of reading anything someone else did, and after a few weeks I had a lower bound of $\sqrt m$ with a short and very clear proof, which, obviously contradicted the upper bound in the thesis.
Now, it is today that I'm a middle age cynical person with fairly low opinion of human abilities and standards of behavior, including my own. At that time I was a young boy to whom my adviser looked if not like a semi-god, then, at least, as an impeccable mathematical professor of intelligence bordering on supernatural. It was absolutely impossible that he could pass anyone with a wrong result!
So, I set up looking for a mistake in my own argument. By the end of the second week of search I was feeling like in a few days I would need to be sent to a mental asylum if I spend just a few more hours on it. The argument was absolutely clean. The heretical thought crossed my mind that the thesis might contain an error, after all, and I started to sift through 50 pages of dense text in which I didn't even know some words. In three more weeks I had read all of it and saw no error either. Back to my proof. Watertight. Back to the thesis and its half-page long computations. Nothing. When I finally found the mistake in the thesis (which was as stupid as $\sin x\to 1$ as $x\to 0$ done en passe in the middle of a long sophisticated limit computation with trigonometric functions), I was half insane.
Moral. If you see an error in a (at least, mathematical) work, by all means, let it be known! It can save if not the life, then the sanity of someone in the future, while all you can harm is something as ephemeral as "self-esteem".
The story is real though I omitted the names (some of which are easy to figure out and some are not).
The best way to report the error is, of course, to figure out what should really be there yourself first and to offer both the criticism and the way out simultaneously. Some good joint papers have been written exactly this way :-)
You should do nothing and hope that the people who find the errors in your own documents are kind enough to do the same. (There is at least one error in my own master's thesis and at least one in my doctoral dissertation. There are also errors in textbooks by respected authors. I've found one in a fifth edition of a book; I checked and it's in the first four editions, too.)
If the document were a book, a web page, or something else amenable to revision, one would send the author a polite note. The "do nothing" advice is for a "one and done" document like a thesis or dissertation.
I think that not finding an error in a thesis would be much more impressive. If you've never seen errors in a scientific piece of work before, it is that you are not very attentive. You probably know of the famous Excel error of Reinhart-Rogoff. Actually, this kind of thing happens all the time, in the best papers too. And it's normal. Scientists are human beings. That is why science is about replication. I don't even speak about typos and such. 90% of scientists writing in English are not English native speakers, so of course they (we) make a lot of mistakes. And so what ? Is it better to spend 100 hours reading again and again the same paper to correct a handful of spelling mistakes without any consequences, or should this time be better employed solving scientific problems ?
There are at least two errors in my thesis. One I corrected in preparing a portion of the thesis for publication. The other remains unresolved, but I believe it can be patched up. I'm sure there are some others as well. It bothers me that they are there, but not to the point that I lose sleep over it. At this point in my career, it is probably almost irrelevant. I work in another area now, but I would like to go back and fix them some time, when I can find the time.
What are you really hoping to accomplish by emailing the person's advisor? It sounds like you want to have the student's "case" re-opened and that you possibly disagree with the person's worthiness of his or her qualification. This is a huge deal (not a small one, by any measure). I think you are opening up the proverbial can of worms, and for no good reason I might add. Sorry to be blunt, but is this your business, really? (The question is only semi-rhetorical, but I think the answer is "no".)
Please read the responses of "Anonymous Mathematician" and Bob Brown above --- they are quite on the mark.
If the author wrote the thesis relatively recently, then do let him or her know directly (not through the advisor), as your constructive criticism could be useful as he or she prepares publications. If the person submitted their thesis some time ago, check to see which publications, if any, arose from it. Maybe the gaps have been closed in those publications.
These things are all that you should do. Leave it alone, otherwise.
A committee of experts at some point decided that the person's thesis was worthy of a degree. They may not have cared that certain gaps existed in the write up. What was presented in the thesis was sufficient and interesting enough for them. The student succeeded in demonstrating a sufficient expertise, by their standards. The thesis may not live up to your own personal standards, but that doesn't matter at all. When you are on a graduate committee, you can apply your standards as you see fit.
If no academic dishonesty has been committed, then I say that you should leave this issue alone. Focus your efforts on your own work and achieving something that meets your own standards.