I search for a module $M$ with its singular submodule $Z(M)$ the empty set, i.e. for every element $m$ of $M$ the annihilator of $m$ in $R$ is not essential, say, as right $R$-module; or proving that such a module does not exist!
Of course, I know that the product $Z(M). soc(R_R)$ is always zero, where $soc(R_R)$ is the socle of $R_R$.
The annihilator of $0\in M$ is $R$, which is always going to be essential. The singular submodule always has at least that.