At several occasions when I was applying for jobs, I found it incredibly helpful to look at application materials from other mathematicians (Ben Webster has an amazing compilation of these). So here are my own application materials: hopefully they can be helpful to someone else, even if only as an example of what *not* to do.
From the 2016-17 search:
Research Statement (no, I didn't end up doing most of this)
Of course, there are many things no one tells you as a student: it is very important to go to conferences, try to give (hopefully good!) talks and just talk to other people. I believe (at least at this stage) these are more important than application materials.
From the 2019-20 search:
From the 2021-22 search:
Research Statement (I am of course still working on problems which are very closely related to this)