  The effects which quantum fields and an $\alpha_0 R^2$ term in the
gravitational Lagrangian have on future singularities are investigated. While
all values of $\alpha_0$ are considered, an emphasis is placed on those values
which are compatible with the universe having undergone Starobinsky inflation
in the past. These are also values which lead to stable solutions to the
semiclassical backreaction equations in the present universe. The dark energy
is modeled as a perfect fluid, and the focus is on type I-IV singularities and
little rips which result when the classical Einstein equations are solved with
various types of dark energy as a source. First, evidence is provided that the
energy densities of massive conformally coupled scalar fields approach that of
the conformally invariant scalar field as a type III singularity is approached.
Then, solutions to the semiclassical backreaction equations are investigated
when conformally invariant fields and the $ \alpha_0 R^2$ term in the
gravitational Lagrangian are present. General proofs regarding the behaviors of
the solutions are given. The proofs are illustrated by analytic and numerical
calculations in specific cases.
