  We systematically analyze the Dirac and the Majorana mass matrices in seesaw
models with two heavy right-handed neutrinos. We perform thorough
classification of the vanishing matrix elements which are compatible with the
results from the current neutrino oscillation experiments. We include the
possibility of a non-diagonal Majorana mass matrix which leads to new solutions
viable with data. In a basis where the Majorana mass matrix is diagonal, these
solutions imply a Dirac matrix with specific relationships amongst its
elements. We find that at the level of total 4 zeros together in the Dirac and
the Majorana sectors, the mass matrices are almost consistent with the data but
one mixing angle is predicted to be unsuitable. At the next level, i.e. with
total 3 zeros, only seven patterns of mass matrices describe the experimental
data well. The seven solutions have testable predictions for the future
neutrino experiments. In particular, each solution has definite predictions
about the observation of the 1-3 leptonic mixing angle and the effective mass
measured in neutrino-less double beta decay. The solutions of the mass matrices
contain novel texture forms and provide new insights into the lepton-generation
structure. We also discuss possible connections between these textures and the
tri-bimaximal mixing to search for symmetry principles behind the mass matrix
structure.
