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For an application cover letter, is it better to sign it with just my name, or with "Ph.D." after my name?
What I am not sure of is that at the beginning of the letter, I do state when and where I got my PhD from. For reference, I am applying for positions in mathematics.
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.
Since I am looking at tenure track cover letters in mathematics anyway, I just opened 12 of them, more or less at random, to gather statistics on this point. Of these letters:
The minority three had at least some of their schooling outside of the US.
Upshot: it is distinctly more common not to sign "PhD", it probably says something about your academic culture, and people play it either way. I have to disagree with @Anonymous Mathematician's assessment that putting a PhD after your signature would be "a bad idea". I have read (literally!) hundreds of cover letters over the last month, and I have never noticed up until now who does this and who doesn't. There are things that appear in cover letters that I find slightly annoying, and this is not one of them. But since you are asking: probably don't do it.
I disagree with the herd, here. When writing formal business letters, your signature block generally contains your name, title, and positions.
It is not "bragging" to include Ph.D. on the line that follows your signature, and formality calls for it. You would not include Ph.D. in your actual signature-- that would look presumptuous.
So far as I know, I can't think of a case where using a formal style in a formal communication would count against you. I can think of many cases where handling a communication that calls for formality in an informal way would be inappropriate.
UPDATE (and feel free to revoke upvotes!): Frankly, the idea that omitting or including your title on a cover letter will have any impact on a search committee is ludicrous. If that's the factor in your application package that makes the difference between an offer and a no offer, your CV must be pretty good.
No. As much as you'd like to have some sort of a "kudos benefit" from having become a Doctor of Philosophy - just don't. They already know you're a Doctor of Philosophy, don't run the risk of sounding like a full-of-themselves Doctor of Philosophy... even if they don't interpret it that way - they might.
