  We study the accelerating cosmology in massive $F(R)$ bigravity via the
reconstruction scheme. The consistent solution of the FRW equations is
presented: it includes Big and Little Rip, quintessence, de Sitter and
decelerating universes described by the physical $g$ metric while the
corresponding solution of the universe described by the reference $f$ metric is
also found. It is demonstrated that in general the cosmological singularities
of $g$ metric are not always manifested as cosmological singularities of the
reference $f$ metric. We give two consistent ways to describe the Big and
Little Rip, quintessence, de Sitter and decelerating universes. In one of the
consistent solutions, the two metrics $g$ and $f$ coincide with each other,
which may indicate the connection with the convenient single metric background
formulation. For another solution, where two metrics $g$ and $f$ do not
coincide with each other, here always appears super-luminal mode.
