Assume I have a singular algebraic surface $X$ over an algebraically closed field (characteristic zero if you want) which is singular in a *finite set of points*. I am looking for a condition as to the nature of these singularities which will guarantee that after blowing up $X$ in each of the singular points **once**, I will get a smooth surface.

In my fever dreams, you find a reference for such a statement from a text book with a very laid-out, comprehensible proof. Any reference, however, is welcome. Thanks!