I would like to know if there can be some kind of classification of normal rational surfaces with Gorenstein singularities, such that their canonical divisor is effective. 


**Additional question.** Are there such surfaces at all?


I could imagine constructing such a surface by blowing up several points on an elliptic curve in $\mathbb CP^2$ and then contracting the proper transform of the curve, but will this give an example? 



A classification of rational surfaces with effective $K$
Here is an example (I hope!).


Take $X$ a double cover of $\mathbb P^2$ branched over a normal sextic $B$. It is a normal Gorenstein surface and the standard formulae for double covers give $K\_X=0$. 
Now assume that $B$ has an ordinary quadruple point $P$ and is smooth elsewhere, so that $X$ has an elliptic Gorenstein singularity at the point $Q$ lying over $P$. 
The minimal desingularization $Y$ of $X$ can be obtained by blowing up the plane at $P$ and taking base change + normalization of the cover. 
So $Y$ is a double cover of $\hat{\mathbb P}^2$ branched on the strict transform $B'$ of $B$. 
The pencil of lines through $P$ induces on $Y$ a linear pencil of rational curves, so $Y$ is rational. 
(Indeed, by the usual formulae for double covers it is easy to show that $\chi(Y)=1$ and that $K\_Y$ is the pull back of $-E$, where $E$ is the exceptional curve of $\hat{\mathbb P}^2\to \mathbb P^2$, and this gives a different proof of the fact that $Y$ is ruled.)


More generally, one can take a curve $B$ of degree $2r$ ($r\ge 3$) with a point $P$ of multiplicity $2r-2$. 


I've no idea whether a classification exists, I had never considered this question before. 


