Sign application cover letter with "PhD" or not?
In the U.S. mathematics departments I'm familiar with, writing "Ph.D." after your name would not be disastrous, but it would be a bad idea.  It can look insecure, like you are worried readers will assume you don't have a Ph.D. if you don't remind them frequently, or pompous, like you feel having a Ph.D. is an important distinction that must be emphasized, and I see no upside to balance the risk of appearing insecure or pompous.

To clarify, I don't think merely listing "Ph.D." after your name would undermine an otherwise great application.  Nobody is going to take it that seriously.  However, it can be tricky to get the tone right in academic job applications, and some applicants inadvertently write things that could be read as insecure or pompous.  (Indeed, it's natural to feel insecure, and it's easy to come across as pompous if you try too hard to convince the reader that you'd make a great hire.)  The danger is that the reader might piece together several small things into an overall negative impression, maybe even subconsciously.  From this perspective, it's safest to eliminate issues you can identify, even if they wouldn't be decisive by themselves, just in case they might reinforce other things you're unaware of.

Note that conventions for the post-nominal use of "Ph.D." may vary between countries or even universities, as well as between fields.  This is a matter of culture, and you'll need to figure out what the culture is like where you are applying.