  In this paper we study the a posteriori bounds for a conforming piecewise
linear finite element approximation of the Signorini problem. We prove new
rigorous a posteriori estimates of residual type in $L^{p}$, for $p \in
(4,\infty)$ in two spatial dimensions. This new analysis treats the positive
and negative parts of the discretisation error separately, requiring a novel
sign- and bound-preserving interpolant, which is shown to have optimal
approximation properties. The estimates rely on the sharp dual stability
results on the problem in $W^{2,(4 - \varepsilon)/3}$ for any $\varepsilon \ll
1$. We summarise extensive numerical experiments aimed at testing the
robustness of the estimator to validate the theory.
