  The $\mu\nu$SSM has been proposed to solve simultaneously the $\mu$-problem
of the MSSM and explain current neutrino data. The model breaks lepton number
as well as R-parity. In this paper we study the phenomenology of this proposal
concentrating on neutrino masses and the decay of the lightest supersymmetric
particle (LSP). At first we investigate in detail the $\mu\nu$SSM with one
generation of singlets, which can explain all neutrino data, once 1-loop
corrections are taken into account. Then we study variations of the model with
more singlets, which can generate all neutrino masses and mixings at
tree-level. We calculate the decay properties of the lightest supersymmetric
particle, assumed to be the lightest neutralino, taking into account all
possible final states. The parameter regions where the LSP decays within the
LHC detectors but with a length large enough to show a displaced vertex are
identified. Decay branching ratios of certain final states show characteristic
correlations with the measured neutrino angles, allowing to test the model at
the LHC. Finally we briefly discuss possible signatures, which allow to
distinguish between different R-parity breaking models.
