  In order to investigate if the anisotropy of the spacetime may induce future
singularities at a finite value of the cosmic time on the evolution of
cosmological models, we study vacuum and non-vacuum Bianchi type I spacetimes
exhibiting such future singularities. We show that in the case of Kasner vacuum
cosmologies the spacetime may rip itself apart in a finite time, and only in
the direction corresponding to the unique scale factor which diverges at this
finite value of the cosmic time. The other two directional scale factors and
the average scale factor do not diverge and tend to zero at this time, while
the directional and average expansion rates also become infinite. Due to the
absence of the matter content this anisotropic future singularity is induced by
the shear scalar, which also blows up at this time. We call such a singularity
"Vacuum Rip". For non-vacuum solutions we discuss fully anisotropic Bianchi
type I spacetimes filled with a stiff fluid and ellipsoidal (axisymmetric)
cosmological models filled with matter with isotropic and anisotropic
barotropic pressure, and characterized by $\sigma/\theta=const$, where $\sigma$
and $\theta$ are the shear scalar and the expansion scalar respectively.
