  We solve massive gravity field equations in the framework of locally
homogenous and vanishing scalar invariant (VSI) Lorentzian spacetimes, which in
three dimensions are the building blocks of constant scalar invariant (CSI)
spacetimes. At first, we provide an exhaustive list of all Lorentzian
three-dimensional homogeneous spaces and then we determine the Petrov type of
the relevant curvature tensors. Among these geometries we determine for which
values of their structure constants they are solutions of the field equations
of massive gravity theories with cosmological constant. The homogeneous
solutions founded are of all various Petrov types : I_C, I_R, II, III, D_t,
D_s, N, O; the VSI geometries which we found are of Petrov type III. The Petrov
types II and III are new explicit CSI spacetimes solutions of these types. We
also examine the conditions under which the obtained anti-de Sitter solutions
are free of tachyonic massive graviton modes.
