  The $\mu\nu$SSM is a simple supersymmetric extension of the Standard Model
(SM) capable of describing neutrino physics in agreement with experiments. We
perform the complete one-loop renormalization of the neutral scalar sector of
the $\mu\nu$SSM with three generation of right-handed neutrinos in a mixed
on-shell/$\overline{\mathrm{DR}}$ scheme. We calculate the full one-loop
corrections to the neutral scalar masses of the $\mu\nu$SSM. The one-loop
contributions are supplemented by available MSSM higher-order corrections. We
obtain numerical results for a SM-like Higgs-boson mass consistent with
experimental bounds, while simultaneously agreeing with neutrino oscillation
data. We illustrate the distinct phenomenology of the $\mu\nu$SSM in scenarios
in which one or more right-handed sneutrinos are lighter than the SM-like Higgs
boson, which might be substantially mixed with them. These scenarios are
experimentally accessible, on the one hand, through direct searches of the
right-handed sneutrinos decaying into SM particles, and on the other hand, via
the measurements of the SM-like Higgs-boson mass and its couplings. In this way
the parameter space of the $\mu\nu$SSM can be probed without the need to
propose model dependent searches at colliders. Finally, we demonstrate how the
$\mu\nu$SSM can simultaneously accommodate two excesses measured at LEP and LHC
at $\sim 96$ GeV at the $1\sigma$ level, while at the same time reproducing
neutrino masses and mixings in agreement with neutrino oscillation
measurements.
