  We construct stationary flat three-dimensional Lorentzian manifolds with
singularities that are obtained from Euclidean surfaces with cone singularities
and closed one-forms on these surfaces. In the application to (2+1)-gravity,
these spacetimes correspond to models containing massive particles with spin.
We analyse their geometrical properties, introduce a generalised notion of
global hyperbolicity and classify all stationary flat spacetimes with
singularities that are globally hyperbolic in that sense. We then apply our
results to (2+1)-gravity and analyse the causality structure of these
spacetimes in terms of measurements by observers. In particular, we derive a
condition on observers that excludes causality violating light signals despite
the presence of closed timelike curves in these spacetimes.
