  We demonstrate long trion spin lifetimes in a WSe$_2$ monolayer of up to 150
ns at 5 K. Applying a transverse magnetic field in time-resolved Kerr-rotation
measurements reveals a complex composition of the spin signal of up to four
distinct components. The Kerr rotation signal can be well described by a model
which includes inhomogeneous spin dephasing and by setting the trion spin
lifetimes to the measured excitonic recombination times extracted from
time-resolved reflectivity measurements. We observe a continuous shift of the
Kerr resonance with the probe energy, which can be explained by an
adsorbate-induced, inhomogeneous potential landscape of the WSe$_2$ flake. A
further indication of extrinsic effects on the spin dynamics is given by a
change of both the trion spin lifetime and the distribution of g-factors over
time. Finally, we detect a Kerr rotation signal from the trion's higher-energy
triplet state when the lower-energy singlet state is optically pumped by
circularly polarized light. We explain this by the formation of dark trion
states, which are also responsible for the observed long trion spin lifetimes.
