Should I use my university's letterhead for a cover letter for a job application? 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.