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We study all four types of finite-time future singularities emerging inlate-time accelerating (effective quintessence/phantom) era from$\mathcal{F}(R,G)$-gravity, where $R$ and $G$ are the Ricci scalar and theGauss-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 thefinite-time future singularities emerge. Furthermore, we discuss a possible wayto cure the finite-time future singularities in $F(G)$-gravity and$\mathcal{F}(R,G)$-gravity by taking into account higher-order curvaturecorrections. The example of non-singular realistic modified Gauss-Bonnetgravity is presented. It turns out that adding such non-singular modifiedgravity to singular Dark Energy makes the combined theory to be non-singularone as well.

Finite-time future singularities in modified Gauss-Bonnet and $\mathcal{F}(R,G)$ gravity and singularity avoidance