Finite-time future singularities in modified Gauss-Bonnet and
  $\mathcal{F}(R,G)$ gravity and singularity avoidance
  We study all four types of finite-time future singularities emerging in
late-time accelerating (effective quintessence/phantom) era from
$\mathcal{F}(R,G)$-gravity, where $R$ and $G$ are the Ricci scalar and the
Gauss-Bonnet invariant, respectively. As an explicit example of
$\mathcal{F}(R,G)$-gravity, we also investigate modified Gauss-Bonnet gravity,
so-called $F(G)$-gravity. In particular, we reconstruct the $F(G)$-gravity and
$\mathcal{F}(R,G)$-gravity models where accelerating cosmologies realizing the
finite-time future singularities emerge. Furthermore, we discuss a possible way
to cure the finite-time future singularities in $F(G)$-gravity and
$\mathcal{F}(R,G)$-gravity by taking into account higher-order curvature
corrections. The example of non-singular realistic modified Gauss-Bonnet
gravity is presented. It turns out that adding such non-singular modified
gravity to singular Dark Energy makes the combined theory to be non-singular
one as well.
