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 nontrivial 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 behavior of local measures of entanglement is fully encompassed in the usual statistical mechanics framework.