  We determine the conjugacy classes of the Poincar\'e group
$\mathrm{ISO}^+(n,1)$ and apply this to classify the stationary trajectories of
Minkowski spacetimes in terms of timelike Killing vectors. Stationary
trajectories are the orbits of timelike Killing vectors and, equivalently, the
solutions to Frenet-Serret equations with constant curvature coefficients. We
extend the $3+1$ Minkowski spacetime Frenet-Serret equations due to Letaw to
Minkowski spacetimes of arbitrary dimension. We present the explicit families
of stationary trajectories in $4+1$ Minkowski spacetime.
