  In the context of Standard Model (SM) extensions, the seesaw mechanism
provides the most natural explanation for the smallness of neutrino masses. In
this work we consider the most economical type-I seesaw realization in which
two right-handed neutrinos are added to the SM field content. For the sake of
predictability, we impose the maximum number of texture zeros in the lepton
Yukawa and mass matrices. All possible patterns are analyzed in the light of
the most recent neutrino oscillation data, and predictions for leptonic CP
violation are presented. We conclude that, in the charged-lepton mass basis,
eight different texture combinations are compatible with neutrino data at
$1\sigma$, all of them for an inverted-hierarchical neutrino mass spectrum.
Four of these cases predict a CP-violating Dirac phase close to $3\pi/2$, which
is around the current best-fit value from global analysis of neutrino
oscillation data. If one further reduces the number of free parameters by
considering three equal elements in the Dirac neutrino Yukawa coupling matrix,
several texture combinations are still compatible with data but only at
$3\sigma$. For all viable textures, the baryon asymmetry of the Universe is
computed in the context of thermal leptogenesis, assuming (mildly) hierarchical
heavy Majorana neutrino masses $M_{1,2}$. It is shown that the flavored regime
is ruled out, while the unflavored one requires $M_{1} \sim 10^{14}$ GeV.
