  We initiate a systematic study of the self-interactions of a massive spin-2
"graviton" consistent with up to $\mathcal{N}=4$ supersymmetry. Using a
recently developed massive on-shell superspace formalism, we construct the most
general set of cubic massive graviton amplitudes in a form with all
supersymmetry and Lorentz invariance manifest. We find that for
$\mathcal{N}\geq 3$ supersymmetry, the family of consistent interactions
coincide with those of the ghost-free dRGT model. For $\mathcal{N}=4$ (maximal)
supersymmetry there is a single consistent cubic interaction which coincides
with the unique structure required for the absence of asymptotic
superluminality. Additionally, we discuss the structure of interactions in the
high-energy limit, connections to supersymmetric Galileons and the possibility
of a supersymmetric massive double copy.
