The title says it all, what's a good dense open of $\bar{M}_g,n(X,\beta)$ which play the role of ${M}_g$ in $\bar{M}_g$?

My first (naive) guess is maps from a genus $g$ smooth curve to $X$ which represents the class $\beta$. But I'm a little bit concerned, is it dense for sure? Could it happen that in some cases one has singular curves only? Can a stable map from a singular curve always deform to a map from a smooth curve of genus $g$?