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.

Comment: 35 pages, 23 figures; minor corrections, final version published in JHEP