  We propose a simple extension of the MSSM based on extra compact dimensions
which includes an $SU(2) \otimes U(1)$ singlet superfield. The fermion present
in this superfield is the sterile neutrino, which combines with one linear
combination of $\nu_e-\nu_{\mu}-\nu_{\tau}$ to form a Dirac pair whose mass
accounts for the LSND anomaly. Its small mass can be ascribed to a volume
suppression factor associated with extra compact dimensions. On the other hand
the sterile neutrino scalar partner can trigger the spontaneous violation of
R-parity, thereby inducing the necessary mass splittings to fit also the solar
and atmospheric neutrino data. Thus the model can explain all neutrino
oscillation data. It leads to four predictions for the neutrino oscillation
parameters and implies that the atmospheric neutrino problem must include at
least some $\nu_{\mu} \to \nu_s$ oscillations, which will be testable in the
near future. Moreover it also predicts that the lightest supersymmetric
particle (LSP) decays visibly via lepton number violating modes, which could be
searched for at present and future accelerators.
