We already know the bound of the order of the finite subgroups of the $Mod(S_g)$. If we take a further step, to find all the finite subgroups, then what is the result for low genus cases? For example, are there some list of all types of the finite subgroups of $Mod(S_g)$ for $g \leq 10$?
As pointed out in the preceding answer, what you are looking for are all possible automorphisms of a compact Riemann surface of genus $g$. You'll find some answers at this question.
Since the maximal order of a finite subgroup of $Mod(S_g)$ is $84(g-1)$ for $g\ge 2$, and this bound is realised for infinitely many $g$'s, listing all finite subgroups might be quite hard for bigger $g$. ( Larsen proved the remarkable result that the frequency of $g$ for which the bound $84(g − 1)$ is attained is the same as the frequency of the perfect cubes in the integers.) Also, all finite groups can be realised as a subgroup of $Mod(S_g)$, for $g\ge 2$.
So it seems to me that such a list is difficult to obtain (I have not seen it), but I am not an expert, and perhaps someone knowns more here. Of course, the bound for finite cyclic subgroups is only $4g+2$, so that a list of finite cyclic subgroups seems much easier to obtain.