Let $X$ be Hausdorff. What are the requirements for $\sim$ that the quotient space $X / \sim$ becomes Hausdorff?
Observation:
If $X / \sim$ is Hausdorff, then all equivalence classes of $\sim$ in $X$ are closed:
If there was a $y \in \overline{[x]} \setminus [x]$, then one could not seperate $[y]$ from $[x]$ in $X / \sim$.
Does the converse hold?
Are there equivalence relations $\sim$ on a Hausdorff space $X$ where the sets $[x]$ are closed, but $X / \sim$ is not Hausdorff?
For that consider the two examples:
The following applet visualizes differerent topologies in $\mathbb{R}^2/\sim$. The orange shape corresponds to an open neighborhood of $[x]$ in the given topology.