  The possibility to have singular accelerated evolution in the context of
$F(R)$ bimetric gravity is investigated. Particularly, we study two singular
models of cosmological evolution, one of which is a singular modified version
of the Starobinsky $R^2$ inflation model. As we demonstrate, for both models in
some cases, the slow-roll parameters become singular at the Type IV
singularity, a fact that we interpret as a dynamical instability of the theory
under study. This dynamically instability may be an indicator of graceful exit
from inflation and we thoroughly discuss this scenario and the interpretation
of the singular slow-roll parameters. Furthermore, it is demonstrated that for
some versions of $F(R)$ bigravity, singular inflation is realized in consistent
way so that inflationary indices are compatible with Planck data. Moreover, we
study the late-time behavior of the two singular models and we show that the
unified description of early and late-time acceleration can be achieved in the
context of bimetric $F(R)$ gravity.
