Distinguishing off-shell supergravities with on-shell physics

We show that it is possible to distinguish between different off-shell completions of supergravity at the on-shell level. We focus on the comparison of the "new-minimal" formulation of off-shell four-dimensional N=1 supergravity with the "old-minimal" formulation. We show that there are three-manifolds which admit supersymmetric compactifications in the new-minimal formulation but which do not admit supersymmetric compactifications in other formulations. Moreover, on manifolds with boundary the new-minimal formulation admits "singleton modes" which are absent in other formulations.