Minimal massive supergravity is the supersymmetric extension of minimal massive gravity in three dimensions. The theory admits a supersymmetric anti-de Sitter vacuum around which the propagating modes of spin 2 and 3/2 combine into a supermultiplet. In this letter, we determine the most general supersymmetric solution of the theory by analyzing its Killing spinor equations. Just as for topologically massive supergravity, the general supersymmetric solution is a plane wave with a null Killing vector field. As a particular subclass we find null-warped AdS3 spaces and, with proper periodic identifications, null-warped AdS3 black holes.