I currently teach at University X and am applying for a job at University Y. Should I write my cover letter on University X's letterhead, or use a blank page?
No. It is you as a person who is applying, not you as a representative for your university. Univ. Y would probably look very negatively on an application on such lerterhead, as would Univ. X. Letterheads are intended for your official business of your position at X as teacher, researcher etc. Applying for a job or for example writing somewhere for your private business is just that, private. You can, however, create your own private letterhead to use for such instances but this should reflect your private standing and show only private address, phone and e-mail.
So keep the official business separate from your private. As alluded to this also applies to e-mail, something most people forget about. It is good to have your own private e-mail for instances such as this.
I personally side with Peter Jansson on this one (do not use an institutional letterhead if you're not conducting business on behalf of your institution), but you will have to note that this position is not universally shared. I cite only one example, of somewhat high-profile blogger/consultant, who says:
Your letter must be on letterhead if you have a current academic affiliation of any kind. This is not negotiable.
I think it may be a field-specific and/or generation issue: some people, and some fields (humanities/law/medicine) have more attachement to older traditions and think a letterhead is a crucial part of correspondence etiquette.
I would ask a senior person in your field. In my experience in mathematics in the US, no one gives a flying flip how you format your cover letter; in all likelihood, no one will read it. Using department stationary is common, though far from universal and I don't think affects anyone's thinking one way or the other.
I don't think this advice is universally applicable; I know in many other disciplines, cover letters are read carefully, and thus their professionalism will have some salutary effect. My personal feeling is that using your current institution's letterhead, your office address, etc. looks more professional, but obviously this isn't a universal feeling, so all the more reason to check about your field specifically.
In academia, if you are applying for an academic position (e.g., assistant adjunct prof applying for assistant prof), regardless of whether it is at the same of different institution, it is absolutely essential that you use the letterhead for the institution with which you are currently affiliated. I realize that this is different for private companies, where "company" letterhead implies official communication on behalf of the "company." But in academia, the letterhead is an indicator that you are recognized as a member of a scholarly community and you have the rights, privileges, and responsibilities associated with your academic appointment. In academia, it does NOT mean you are conducting official institutional business. Official institutional business will have "Office of the Chancelor" in the letterhead or "Office of XYZ Department Chair." If the chancellor or department chair were to apply for jobs, they would probably still use institutional letterhead but not have this "Office of the __" line in the letterhead.
Using your institutional letterhead is ABSOLUTELY ESSENTIAL when applying for academic jobs. Even (especially) if you are a grad student. Everyone knows that you are not speaking on behalf of the institute, the letterhead is an indication of your membership in that scholarly community.
One point that I'm missing so far, but which IMHO makes quite some difference is the position which you are in:
My "default" opionon on this question is not to use your employer's letterhead, nor your email at your employer's.
The reason is that unless your relationship with your current employer is really bad (in which case you don't want to use their letterhead), (ab)using your employer's letterhead demonstrates illoyality: not only are you not acting in as an official of your employer, but as @scottishwildcat already pointed out, you are presumably acting against your employer's interests.
But in academia, there are certain situations where your old university is anywhere between quite happy with over positively encouraging you to expecting you to apply for another job.
You may be in a stage of your career where a change of university is expected or at least reasonable and you and your employer agree on this (close to finishing your degree, did a postdoc abroad but want to move home again, want to move for family reasons, want to become a professor which in some countries you cannot at your "home" university, ...), or you are on a project position and for external reasons they cannot keep you.
In this case, IMHO you may use the university letterhead (although I'd probably still not do it). But if you do so, you should make sure the university where you apply knows unambiguously that your university is happy with your application for their position, e.g. by naming your current supervisor as reference.
As for the email, free emails are available also with sober username and sober providers, so that shouldn't be a problem, neither.
Do whatever you want.
There are strong and conflicting opinions on this ("it would set off a red flag" to not use institutional letterhead, etc. vs "I put those letters at the bottom of the pile.")
We should ask ourselves, as academics: if we are evaluating people's job applications using such incredibly fine distinctions in academic etiquette, either:
1) we are prioritizing completely useless information and probably introducing a good deal of bias against folks like international students and first-gen college students along with it!
2) academia is so irredeemably petty that getting this right is actually an important sign of success in an academic position.
Either answer doesn't make academia sound like a place you'd want to work. I have faith that people are not actually making such important decisions using trivia. I therefore suggest that the original poster should choose whichever option allows them to make the content of their letter clearer, i.e., if you need the space for more information, don't use the letterhead!
I fully understand the notion of not writing on letterhead except in an official capacity, but consider the tradition of using a hotel's letterhead when staying with them as a guest. I appreciate this is perhaps almost akin to sending a picture postcard (and an advertising opportunity for the hotel), yet this clearly is not a matter of the hotel's business activity.
This practice extended to the English with houses large enough to have guest rooms, where the etiquette is to provide headed notepaper of your own for your guests to use.
Completely agree that you should not use letterhead from the academic institution you are trying to leave. I put those letters at the bottom of the pile when reviewing job applications for teaching positions. Only caveat is when you hold a postdoc, but if you are tenure track or more, then no, use your own.