For complex projective smooth surface $X$, let $M$
be the coarse moduli scheme of rank-two stable sheaves with fixed Chern classes.
Grasping the birational structure of $M$, for example its Kodaira dimension, is a fundamental problem.
However, in the case where $\kappa(X)>0$, the study of this problem has not necessarily been active in recent years.
In this article we survey the study of this problem, especially for the case where $\kappa(X)=1$ and $c_1=0$.
We will also survey some research on the structure of singularities of $M$, and
a minimal model program of $M$.
While explaining motivations, we raise several unsolved problems.
