  We find an explicit cosmological model which allows a special type of
cosmological singularity which we call a $w$-singularity. This singularity has
the scale factor finite, the energy density and pressure vanishing, and the
only singular behaviour appears in a time-dependent barotropic index $w(t)$. It
is different from the type IV cosmological singularity in that it does not
exhibit the divergence of the higher derivatives of the Hubble parameter and
from the big-brake since it does not fulfill the anti-Chaplygin gas equation of
state. We also find an interesting duality between the $w$-singularities and
the big-bang singularities. Physical examples of $w$-singularities appear in
$f(R)$, scalar field and brane cosmologies.
