Let $X$ be a second countable locally compact Hausdorff space and $q: X\to Y$ a quotient map with $Y$ Hausdorff. Suppose that the union of the set of fibres $q^{-1}(y)$ which are singletons contains a dense $G_{\delta}$ in $X$. My question is: what constraints does this place on $Y$?
For example, if $Y$ is countable, then the set of fibres is countable, and each fibre is a closed set, so by the Baire property for $X$, some of the fibres must have non-empty interior. But then the dense-singletons assumption means that some of these fibres with non-empty interior must be singletons, and hence isolated points. Thus $Y$ has isolated points. So $Y$ cannot be the Arhangelskii-Franklin space $S_{\omega}$, which is a quotient of a countable, second countable locally compact Hausdorff space but has no isolated points.
More generally, any countable open subset of $Y$ must contain isolated points, by similar reasoning. Are there any other constraints on $Y$?
(Perhaps I should mention that I do have an example with $Y$ nowhere locally compact. This is based on identifying a countable family of points of $[0,1]$ with a countable dense family in the open unit square $(0,1)\times (0,1)$).