Let $M$ be a smooth manifold, of dimension $n$. I know that there are many types of geometric structures on $M$. A large number of them are captured by the notion of a $G$-structure, which is a reduction of $M$'s frame bundle $F(M)$ to a Lie subgroup $G$ of $GL(n,\mathbb{R})$, the structure group of $F(M)$.

Kobayashi, in his text "Transformation Groups in Differential Geometry," is kind enough to provide a list of some $G$-structures. His list includes structures with which most people are acquainted, such as $O(n)$-structures (Riemannian metrics), $O(n)\times \mathbb{R}_{>0}$-structures (conformal structures), and $SL(n,\mathbb{R})$-structures (volume forms). Several important ones are missing, such as almost CR-structures, which in the case $n=2k+1$ are reductions of $F(M)$ to the subgroup $G_0$ of $GL(2k+1,\mathbb{R})$ with elements $$ \begin{bmatrix} A & x\\ 0 & y \end{bmatrix}, $$ where $A\in GL(k,\mathbb{C})$, $x\in \mathbb{R}^{2k}$, and $y\in \mathbb{R}^\times$.

Questions: Is there a more comprehensive list of $G$-structures somewhere? Are there "exotic" $G$-structures that have appeared in the literature?

veryexotic but popular structure is the $G_2$-structure on $7$-manifolds. $\endgroup$ – Alex Degtyarev Jan 29 '15 at 21:57somereductions of $SO(2n+1)$ to $SO(2n)$ are called contact structures and are indeed studied a lot. Although I do agree that not everything is studied/worth studying, but this does not seem to be relevant to your question title :) $\endgroup$ – Alex Degtyarev Jan 29 '15 at 23:10subgroupis basically a question about the holonomy group of the metric, and I think that such thingsarestudied. $\endgroup$ – Alex Degtyarev Jan 29 '15 at 23:127more comments