We consider a model of neutrino mass based on $R$-parity-violating supersymmetry, with three ${\ensuremath{\mu}}_{i}$, relevant for bilinear $R$-parity-violating terms, and three ${\ensuremath{\lambda}}_{ijk}^{\ensuremath{'}}$, relevant for the trilinear terms. The present neutrino data, after a precise determination of the mixing angle ${\ensuremath{\theta}}_{13}$, severely constrain such models. We make a thorough study of one such class of models that may have interesting signatures at the upgraded LHC. In this class of models, the relevant trilinear couplings are of the form ${\ensuremath{\lambda}}_{i33}^{\ensuremath{'}}$, so if the lighter stop squark ${\stackrel{\texttildelow{}}{t}}_{1}$ is the lightest supersymmetric particle, it will decay only through these couplings, giving rise to events with isolated hard leptons and jets. Even when ${\stackrel{\texttildelow{}}{t}}_{1}$ is the next-to-lightest supersymmetric particle, it can decay via the tiny ${\ensuremath{\lambda}}^{\ensuremath{'}}$ couplings allowed by the neutrino data, although it may face stiff competition from some $R$-parity-conserving decay modes. Using a simple Pythia based simulation, we find that in both the cases the signal consisting of a pair of oppositely charged leptons ($ee$, $\ensuremath{\mu}\ensuremath{\mu}$ or $e\ensuremath{\mu}$) plus jets may be observable at the upgraded LHC experiments for a reasonable range of the ${\stackrel{\texttildelow{}}{t}}_{1}$ mass.