Physical Danger
With the exception of thinking headaches (we get those, don't you?), the field of mathematics is one of the safest careers to pursue. Considering most positions involve a desk at an office, the most dangerous part of your day will probably be the drive to and from work.
The hazards do grow, however, depending on the research subject and how hands-on the mathematician is with their subject. As an applied mathematician working in technology, you're likely to work at some point with hydraulics, electrics, power tools, and/or really sharp drafting compasses (among other things).
Everything more complicated than a piece of paper and pen will offer its own set of dangers. The best thing to do is wear proper safety equipment whenever the situation calls. As a mathematician, you'll be familiar with numerous rules and laws; keep Murphy's Law in mind, and you should do fine.