  We discuss on general grounds some local indicators of entanglement, that
have been proposed recently for the study and classification of quantum phase
transitions. In particular, we focus on the capability of entanglement in
detecting quantum critical points and related exponents. We show that the
singularities observed in all local measures of entanglement are a consequence
of the scaling hypothesis. In particular, as every non-trivial local observable
is expected to be singular at criticality, we single out the most relevant one
(in the renormalization group sense) as the best-suited for finite-size scaling
analysis. The proposed method is checked on a couple of one-dimensional spin
systems. The present analysis shows that the singular behaviour of local
measures of entanglement is fully encompassed in the usual statistical
mechanics framework.
