  We investigate in which cases a singular evolution with a singularity of Type
IV, can be consistently incorporated in deformations of the $R^2$ inflationary
potential. After demonstrating the difficulties that the single scalar field
description is confronted with, we use a general two scalar fields model
without other matter fluids, to describe the Type IV singular evolution, with
one of the two scalar fields being canonical. By appropriately choosing the
non-canonical scalar field, we show that the canonical scalar field corresponds
to a potential that is nearly the $R^2$ inflation potential. If the Type IV
singularity occurs at the end of inflation, the Universe's dynamical evolution
near inflation is determined effectively by the canonical scalar field and at
late-time the evolution is effectively determined by the non-canonical scalar.
We also discuss the evolution of the Universe in terms of the effective
equation of state and we show that the Type IV singularity, that occurs at the
end of inflation, drives late-time acceleration. If however the singularity
occurs at late-time, this might affect the inflationary era. We also
investigate which Jordan frame pure $F(R)$ gravity corresponds to the nearly
$R^2$ inflation scalar potentials we found. The stability of the solutions in
the two scalar fields case is also studied and also we investigate how Type IV
singularities can be incorporated in certain limiting cases of $R+R^p$ gravity
in the Einstein frame. Finally, we briefly discuss a physical appealing
scenario triggered by instabilities in the dynamical system that describes the
evolution of the scalar fields.
