The aim of the paper is to investigate some local properties of the weak congruence lattice of an algebra, which is supposed to possess the constant $\mathbf 0$, or a nullary term operation. Lattice identities are restricted to the zero blocks of weak congruences. In this way, a local version of the CEP, and local modularity and distributivity of the weak congruence lattices are characterized. In addition, local satisfaction by weak congruences of some general lattice identities is proved.