What non-research activities are most valuable when applying for faculty positions in mathematics? I am asking this question for two reasons: first to work on activities to strengthen my CV for the future, and second to write a plan for professional development activities as part of a grant proposal. 

What non-research related activities are most valuable to have on a CV when applying to permanent faculty positions at research-focused universities? For example, how valuable is it to have experience working with graduate students, organizing conferences/seminars, etc.?

I am a pure mathematician, hoping for advice that can apply to both European and American jobs. I currently am working on my second postdoc, although my first one was only six months long.  I had to take some adjunct teaching positions here and there as well, so I have a decent amount of undergraduate teaching experience. I have a limited publication record however (one thing I know I need to work on).
What is valued around here is to show independent work (not just papers with your advisor, have your own research grants). Show involvement with the relevant community (organize conferences, attend conferences, get invited to talk about your work).