  An alternative to the Big Bang cosmologies is obtained by the Big Bounce
cosmologies. In this paper, we study a bounce cosmology with a Type IV
singularity occurring at the bouncing point, in the context of $F(R)$ modified
gravity. We investigate the evolution of the Hubble radius and we examine the
issue of primordial cosmological perturbations in detail. As we demonstrate,
for the singular bounce, the primordial perturbations originating from the
cosmological era near the bounce, do not produce a scale invariant spectrum and
also the short wavelength modes, after these exit the horizon, do not freeze,
but grow linearly with time. After presenting the cosmological perturbations
study, we discuss the viability of the singular bounce model, and our results
indicate that the singular bounce must be combined with another cosmological
scenario, or should be modified appropriately, in order that it leads to a
viable cosmology. The study of the slow-roll parameters leads to the same
result, indicating the singular bounce theory is unstable at the singularity
point, for certain values of the parameters. We also conformally transform the
Jordan frame singular bounce, and as we demonstrate, the Einstein frame metric
leads to a Big Rip singularity. Therefore, the Type IV singularity in the
Jordan frame, becomes a Big Rip singularity in the Einstein frame. Finally, we
briefly study a generalized singular cosmological model, which contains two
Type IV singularities, with quite appealing features.
