  Assuming Majorana nature of neutrinos, we investigate the singular one
texture zero neutrino mass matrices in the flavor basis. We find that for the
normal mass ordering with $m_1=0$, all the six one texture zero classes are now
ruled out at 3$\sigma$ confidence level, whereas for inverted mass ordering
with $m_3=0$ only four classes out of total six can accommodate the latest
neutrino oscillation data at 3$\sigma$ confidence level. Moreover, only two
classes can accommodate the present data at 1$\sigma$ confidence level. We
examine the phenomenological implications of the allowed classes for the
effective Majorana mass, Dirac and Majorana CP-violating phases. Working within
the framework of type-I seesaw mechanism, we present simple discrete Abelian
symmetry models leading to all the phenomenologically allowed classes.
