Let $(R,\mm)$ be a regular local ring or a polynomial ring over a field, and let $I$ be an ideal of $R$ which we assume to be graded if $R$ is a polynomial ring. Let $\astab(I)$, $\overline{\astab}(I)$ and $\dstab(I)$, respectively, be the smallest integer $n$ for which $\Ass(I^n)$, $\Ass(\overline{I^n})$ and $\depth(I^n)$ stabilize. Here $\overline{I^n}$ denotes the integral closure of $I^n$. We show that $\astab(I)=\overline{\astab}(I)=\dstab(I)$ if $\dim R\leq 2$, while already in dimension $3$, $\astab(I)$ and $\overline{\astab}(I)$ may differ by any amount. Moreover, we show that if $\dim R=4$, then there exist ideals $I$ and $J$ such that for any positive integer $c$ one has $\astab(I)-\dstab(I)\geq c$ and $\dstab(J)-\astab(J)\geq c$.
