I am looking for an example of a module $M$, a ring $A$, and a prime ideal $\mathfrak p$ such that $\operatorname{depth}_{\mathfrak p} M < \operatorname{depth}_{A_{\mathfrak p}} M_{\mathfrak p}$. How can I find such an example?
If $A$ is local with maximal ideal $\mathfrak m$, and $M = A$, then if $\mathfrak m$ is associated we have $\operatorname{depth}_A M=0$. On the other hand, if I can choose a prime $\mathfrak p$ that is not an associated prime of $A$, we get $\operatorname{depth}_{A_{\mathfrak p}} M_{\mathfrak p} \ge 1$. How can I look for this example?
Let's take $A = k[x,y]/(xy,y^2)$. Then $\operatorname{depth}_A A = 0$. Take $\mathfrak p = (y)$. Is it true however that $\operatorname{depth}_{A_{\mathfrak p}} A_{\mathfrak p} \geq 1$?
Hint. Try a similar example for $A$, that is, a quotient ring of a polynomial ring in three indeterminates.