  As a consequence of the LSND anomaly and other hints of an eV scale sterile
neutrino from particle physics and cosmology, the neutrino sector of the
standard model of particle physics has to be extended and the smallest
extension is the (3+1) model, i.e. three active neutrinos plus one sterile one.
In this work we study the neutrino mass matrix $M_\nu$ with three texture zeros
in the (3+1) model, assuming all the neutrino states are of Majorana type. With
the mass hierarchy between the active and sterile neutrinos as well as the
smallness of the reactor neutrino mixing and active-sterile mixing, the
analytical expressions can be greatly simplified. We systematically examine all
the 120 texture zeros, via both analytical and numerical analysis, and find
that the 100 textures with zeros in the fourth column and row of $M_\nu$ can
hardly be compatible with experiments whereas 19 of the other 20 are feasible.
We can foresee that the active neutrino masses are required to be around 0.01
eV and the next generation neutrinoless double beta decay experiments are
promising to test these textures. Many other phenomenologically meaningful
predictions are also obtained. All the CP conserving (3+1) textures with three
zeros are excluded by experiments.
