Consider the double cover $\pi:S^1 \rightarrow S^1, z \mapsto z^2$ and the pushforward of the constant sheaf $\pi\_{\*}\mathbb{Z}$. This is a locally constant sheaf of rank 2, but not constant (since the space of global sections is rank 1). 


Question: if I choose a basis $u,v$ for the stalk at some point $p$, how to compute the monodromy matrix with respect to this matrix? 



Computing an example of monodromy
It looks like this was asked and answered a while ago. But since it floated to the top
again, let me give another answer. 


First in classical language, the monodromy exchanges
sheets $ \sqrt{z}\leftrightarrow -\sqrt{z}$. If you imagine taking formal linear combinations
of these, then you should be able to recover the monodromy matrix given
in Francesco's answer.


Here is a more abstract general point of view.
If $Y$ is a connected (sufficiently nice)
space, the category of locally constant sheaves is equivalent to representations
of $\pi\_1(Y)$ via monodromy. The pullback of a locally constant sheaf
along a covering space $\pi:X\to Y$ corresponds to
restriction from $\pi\_1(Y)$ to $\pi\_1(X)$. Pushforward would be the right adjoint which 
corresponds to induction in the opposite direction. So in particular $\pi\_\*\mathbb{Z}$ is the regular representation of 
$\pi\_1(Y)\to Aut(\mathbb{Z}[G])$ when $\pi$ is Galois with group $G$.


