  Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a
field $K$ and $I\subset S$ a squarefree monomial ideal. In the present paper we
are interested in the monomials $u \in S$ belonging to the socle
$\Soc(S/I^{k})$ of $S/I^{k}$, i.e., $u \not\in I^{k}$ and $ux_{i} \in I^{k}$
for $1 \leq i \leq n$. We prove that if a monomial $x_1^{a_1}\cdots x_n^{a_n}$
belongs to $\Soc(S/I^{k})$, then $a_i\leq k-1$ for all $1 \leq i \leq n$. We
then discuss squarefree monomial ideals $I \subset S$ for which $x_{[n]}^{k-1}
\in \Soc(S/I^{k})$, where $x_{[n]} = x_{1}x_{2}\cdots x_{n}$. Furthermore, we
give a combinatorial characterization of finite graphs $G$ on $[n] = \{1,
\ldots, n\}$ for which $\depth S/(I_{G})^{2}=0$, where $I_{G}$ is the edge
ideal of $G$.
