  Let $(R,\mathfrak{m})$ 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$(I)$ resp. $\overline{\rm astab}(I)$ be the smallest
integer $n$ for which Ass$(I^n)$ resp. Ass$(\overline{I^n})$ stabilize, and
dstab$(I)$ be the smallest integer $n$ for which depth$(I^n)$ stabilizes. Here
$\overline{I^n}$ denotes the integral closure of $I^n$. We show that
astab$(I)=\overline{\rm astab}(I)={\rm dstab}(I)$ if dim$\,R\leq 2$, while
already in dimension $3$, astab$(I)$ and $\overline{\rm astab}(I)$ 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 ${\rm astab}(I)-{\rm
dstab}(I)\geq c$ and ${\rm dstab}(J)-{\rm astab}(J)\geq c$.
