  We demonstrate that the intermediate inflation scenario, is a singular
inflation cosmology, with the singularity at the origin $t=0$ being a pressure
and energy density singularity and particularly a Type III singularity. Also,
we show that the expanding phase of a singular bounce, can be identical to the
intermediate inflation scenario, if the singular bounce has a Type III
singularity at the origin. For the intermediate inflation scenario we examine
the cosmological implications on the power spectrum in the context of various
forms of modified gravity. Particularly we calculate the power spectrum in the
context of $F(R)$, $F(G)$ Gauss-Bonnet gravity and also for $F(T)$ gravity and
we discuss the viability of each scenario by comparing the resulting spectral
index with the latest observational data.
