Let $(R,\mm)$ be a commutative Noetherian ring and $I$ be an ideal of $R$. Brodmann \cite{B1} proved that the set of associated prime ideals $\Ass(I^k)$ stabilizes. In other words, there exists an integer $k_0$ such that $\Ass(I^k)=\Ass(I^{k_0})$ for all $k\geq k_0$. The smallest such integer $k_0$ is called the {\em index of Ass-stability} of $I$, and denoted by $\astab(I)$. Moreover, $\Ass(I^{k_0})$ is called the {\em stable set of associated prime ideals} of $I$. It is denoted by $\Ass^{\infty}(I)$. For the integral closures $\overline{I^k}$ of the powers of $I$, McAdam and Eakin \cite{Me} showed that $\Ass({\overline{I^k}})$ stabilizes as well. We denote the index of stability for the integral closures of the powers of $I$ by $\overline{\astab}(I)$, and denote its stable set of associated prime ideals by $\overline{\Ass}^{\infty}(I)$. Brodmann \cite{B} also showed that $\depth R/I^k$ stabilizes. The smallest power of $I$ for which depth stabilizes is denoted by $\dstab(I)$. This stable depth is called the {\em limit depth} of $I$, and is denoted by $\lim_{k\to\infty}\depth R/I^k$. These indices of stability have been studied and compared to some extend in \cite{Hq} and \cite{Hrv}. The purpose of this work is to compare once again these stability indices. The main result is that if $(R,\mm)$ is a regular local ring with $\dim R\leq 2$, then all 3 stability indices are equal, but if $\dim R=3$, then we still have $\astab(I)=\dstab(I)$, while $\astab(I)$ and $\overline{\astab}(I)$ may differ by any amount. On the other hand, if $\dim R\geq 4$, we will show by examples that in general a comparison between these stability indices is no longer possible. In other words, any inequality between these invariants may occur. Quite often, but not always, $\depth(R/I^k)$ is a non-increasing function on $n$. In the last section we prove that if $(R,\mm)$ is a $3$-dimensional regular local ring and $I$ satisfies $I^{k+1}:I=I^k$ for all $k$, then $\depth R/I^k$ is non-increasing
