  Texture zeroes in neutrino mass matrix $M_\nu$ may give us hints about the
symmetries involved in neutrino mass generation. We examine the viability of
such texture zeroes in a model independent way through a bottom-up approach.
Using constraints from the neutrino oscillation data, we develop an analytic
framework that can identify these symmetries and quantify deviations from them.
We analyze the textures of $M_\nu$ as well as those of $M_M$, the mass matrix
of heavy Majorana neutrinos in the context of Type-I seesaw. We point out how
the viability of textures depends on the absolute neutrino mass scale, the
neutrino mass ordering and the mixing angle $\theta_{13}$. We also examine the
compatibility of discrete flavor symmetries like $\mu$--$\tau$ exchange and
$S_3$ permutation with the current data. We show that the $\mu-\tau$ exchange
symmetry for $M_\nu$ can be satisfied for any value of the absolute neutrino
mass, but for $M_\nu$ to satisfy the $S_3$ symmetry, neutrino masses have to be
quasi-degenerate. On the other hand, both these symmetries are currently
allowed for $M_M$ for all values of absolute neutrino mass and both mass
orderings.
