Application materials
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: (as a result of this I was a postdoc at UC Davis and received at least one more postdoc offer)
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: (as a result of this, I was a postdoc at MPIM)
From the 2021-22 search: (as a result of this I have my current job at UNAM, and received at least one more TT offer)
Research Statement (I am of course still working on problems which are very closely related to this)