Set of All Points at which $X$ is Locally Connected

Let

$$X = \{0\} \times\mathbb{R} \cup \bigcup_{n = 1}^{\infty} \{ (x,y) : (x-n)^2 + y^2 = n^2\},$$

in the subspace topology induced by $\mathbb{R}^2$.

Then $X$ is locally connected at $(0,0)$ - every $B_r((0,0)) \cap X$ for $r > 0$ is connected - but $X$ is not locally connected at any $(0,y)$ with $y \neq 0$, so $S$ is not open.

Generally, $S$ being open alone doesn't tell us much about $X$. $S$ is of course locally connected then, and the exterior of $S$ is nowhere locally connected, but both of these can happen in many ways.

Would this tell us much about the space $X$?

It can tell us something about nice embeddings of $S$, and by extension, something about $X$.

If $X$ is compact and $S$ is open, then shrinking $X\setminus S$ to a single point $\infty$ produces a compact locally connected space $S\cup \{\infty\}$. In particular, $X$ maps onto a locally connected space via a continuous extension of the identity on $S$.

This is a very nice result if you are interested in compactifications.

Theorem 4.1 in this paper also says that $X$ is locally connected if $\overline S=X$ and $X\setminus S$ is totally disconnected. For instance, consider $X=[0,1]$, $S=(0,1)$, and $X\setminus S=\{0,1\}$. Compare to the topologists sine curve.