Let $R$ be a ring with identity, and $Z(R_R)$ be the singular ideal. Is it true that any nilpotent ideal of $R$ lies in $Z(R_R)$? It is well known that any central nilpotent element would belong to the singular ideal.
I assume that $I$ is a nilpotent ideal: $I^n=0$, and $x\in I$. I try to prove that the right annihilator $\operatorname{ann}_r(x)$ of $x$ is essential in the module $R_R$. So, choose $0≠y\in R$. One should look for an $r\in R$ with $yr≠ 0$ and $xyr=0$. Since the element $x\in I$ is nilpotent it is a left zero-divisor. If $xy=0$ we could set $r=1$. But, henceforth I got stuck. Is there any complete proof or counterexample for the raised question?
It is enough to find a right nonsingular ring that isn't semiprime.
Let $R=\left[\begin{smallmatrix}F&F\\0&F\end{smallmatrix}\right]$, the $2\times 2$ upper triangular matrices over a field.
Then $I=\left[\begin{smallmatrix}0&F\\0&0\end{smallmatrix}\right]$ satisfies $I^2=0$, but $r(\left[\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right])=\left[\begin{smallmatrix}F&F\\0&0\end{smallmatrix}\right]=\left[\begin{smallmatrix}1&0\\0&0\end{smallmatrix}\right]R$, but this last thing obviously has complement $\left[\begin{smallmatrix}0&0\\0&1\end{smallmatrix}\right]R$ in $R$, so it is not an essential right ideal. Ergo, $I\nsubseteq Z(R_R)$.
For more nonsingular rings which aren't semiprime, you can perform a search at DaRT.