  We carry out a complete and systematic study of all the possible one-zero
textures for the neutrino mass matrix in light of the recent neutrino
oscillation data which hint to a relatively large non-vanishing value for the
smallest mixing angle ($\theta_z$). We find that all the six possible one-zero
textures are able to accommodate the experimental data, with five cases
allowing also for non-invertible mass matrices. We present symmetry
realizations for all the one-zero singular and non-singular patterns in type-I
and type-II seesaw mechanism schemes.
