I have to find an example of Gorenstein ring which is not local regular.
I take $A=K[|X,Y|]/(XY)$.
This is a local ring. As $XY\in (X,Y)^2$ it results that $A$ is not regular.
$K[|X,Y|]$ is Cohen-Macaulay $\Rightarrow$ $K[|X,Y|]$ is Gorenstein and $XY$ is a nonzero divisor on $K[|X,Y|]$ $\Rightarrow$ $A$ is Gorenstein.
Is it right?