All of them can be described as follows. Let $(A,m)$ be any local ring (commutative). By going modulo $m^3$ we may assume that $m^3=0$. Now $m^2$ is a vector space over $k=A/m$ and let $I\subset m^2$ be any codimension one $k$-subspace. Then $I$ is an ideal in $A$ and $A/I$ will have the property you need and any such looks like this.