  We consider the cosmological evolution of a flat anisotropic Universe in
$f(T)$ gravity in the presence of a perfect fluid. It is shown that the matter
content of the Universe has a significant impact of the nature of a
cosmological singularity in the model studied. Depending on the parameters of
the $f(T)$ function and the equation of state of the perfect fluid in question
the well-known Kasner regime of general relativity can be replaced by a new
anisotropic solution, or by an isotropic regime, or the cosmological
singularity changes its nature to a non-standard one with a finite values of
Hubble parameters. Six possible scenarios of the cosmological evolution for the
model studied have been found numerically.
