Let $X$ be a topological space, $\pi:E\to X$ a fiber bundle over $X$ with fiber $F$ and structure group $G$. Let $\mathcal{F}$ denote the sheaf of continuous sections of the bundle. I probably want to assume $X$ has some reasonable topological properties.
Is there any nice description for the étalé space of $\mathcal{F}$? When $F$ is discrete, i.e. $E$ is a covering space, I would say E itself can be chosen as étalé space. But how about "bigger" F?
The étalé space is an object $\pi:E\to X$ of the slice category $\mathrm{Top}_{/X}$, where $X$ is an object of $\mathrm{Top}$, such that $\pi$ is a local homeomorphism. See that there is an equivalence of categories between $\mathrm{Etale}_{/X}$ of étalé spaces over $X$ and the category $\mathrm{Shv}_{X}$ of sheaves on $X$. A nice way of seeing this is by noting that there is an isomorphism between the sheaf of sections of $\mathrm{Etale}(\mathscr{F})$ and $\mathscr{F}$ itself, and is a form of sheafification. See page $50$ of these notes and Planetmath.