Assuming you've defined "oscillation at a point correctly" (I have not tried to proof-read your definitions), the oscillation function is upper semicontinuous. Thus, you can try googling "oscillation" along with the phrase "upper semicontinuous".
The characteristic function of a Cantor set with positive measure shows that the oscillation function can be discontinuous on a set of positive measure.
On the other hand, because the oscillation function is upper semicontinuous (indeed, being a Baire one function suffices), the oscillation function will be continuous on a co-meager set (i.e. at every point in a set whose complement has first Baire category). Because the set of discontinuities of any function is an $F_{\sigma}$ set, the discontinuities of the oscillation function will be an $F_{\sigma}$ set. Putting the last two results together tells us that the oscillation function always has an $F_{\sigma}$ meager (i.e. first Baire category) discontinuity set. I believe this result is sharp in the sense that given any $F_{\sigma}$ meager set $D,$ there exists a function $f:{\mathbb R} \rightarrow {\mathbb R}$ such that $D$ is equal to the set of all the points at which the oscillation function ${\omega}_{f}$ is not continuous. I don't have time to look into this now, but I believe this sharp result follows from the more precise results proved in [1] and [2] (see also [3]). Regarding what possibilities exist for sets that are $F_{\sigma}$ and meager, see #1-7 in my answer to the math StackExchange question How discontinuous can a derivative be?.
[1] Zbigniew Grande, Quelques remarques sur la semi-continuité supérieure, Fundamenta Mathematicae 126 #1 (1985), 1-13.
[2] Tomasz Natkaniec, On semicontinuity points, Real Analysis Exchange 9 #1 (1983-1984), 215-232.
[3] Janina Ewert, On points of lower and upper semicontinuity of multivalued maps, Mathematical Chronicle 20 (1991), 85-88.
(ADDED NEXT DAY) My conjecture above (where I said I believe this result is sharp in the sense that …) appears to be correct. Indeed, I essentially said as much in this 29 April 2002 sci.math post, where I mentioned that a special case of Theorem 5(a) on p. 561 of [4] (reference below) implies that for each locally bounded non-negative upper semi-continuous function $f:{\mathbb R} \rightarrow {\mathbb R},$ there exists a function $F:{\mathbb R} \rightarrow {\mathbb R}$ such that ${\omega}_{F} = f$. (In analogy with the Fundamental Theorem of Calculus, any such function $F$ is called an ${\omega}$-primitive of $f.)$ It is reasonably well known that any $F_{\sigma}$ meager set can be the discontinuity set for an upper semi-continuous function, and the standard proof of this (see the proof sketch below) gives a function that is also locally bounded and non-negative.
Proof Sketch: Let $D$ be an $F_{\sigma}$ meager subset of ${\mathbb R}.$ Express $D$ as the union of a countable (possibly finite) collection $\{P_n\}$ of closed nowhere dense sets and define $f = \sum \left( 2^{-n}\cdot{\chi}_{P_n}\right),$ where ${\chi}_{P_n}$ is the characteristic function of $P_n$ (i.e. ${\chi}_{P_n}(x) = 1$ if $x \in P_{n},$ and ${\chi}_{P_n}(x) = 0$ if $x \notin P_{n}).$ Then $f$ is a bounded non-negative upper semi-continuous function whose discontinuity set is equal to $D.$
Below are some additional references related to this topic. There are other papers not included, and these can be found by looking for papers by these authors and performing a google phrase search for the title of Kostyrko's paper, "Some properties of oscillation".
[4] Pavel Kostyrko, Some properties of oscillation, Mathematica Slovaca 30 #2 (1980), 157-162.
[5] Zbigniew Duszyński, Zbigniew Jan Grande, and Stanislaw Petrovich Ponomarev, On the $\omega$-primitive, Mathematica Slovaca 51 #4 (2001), 469-476.
[6] Janina Ewert and Stanislaw Petrovich Ponomarev, Oscillation and $\omega$-primitives, Real Analysis Exchange 26 #2 (2000-2001), 687-702.
[7] Cristina Di Bari and Calogero Vetro, Primitive rispetto all'oscillazione [Primitives with respect to oscillation], Rendiconti del Circolo Matematico di Palermo (2) 51 #1 (2002), 175-178.
[8] Janina Ewert and Stanislaw Petrovich Ponomarev, On the existence of $\omega$-primitives on arbitrary metric spaces, Mathematica Slovaca 53 #1 (2003), 51-57.
[9] Stanisław Kowalczyk, On the $\omega$-problem, Real Analysis Exchange Summer Symposium 2011, 120-122.
Best Answer
I try an answer.
Suppose that $f(x)=d(x,\Omega)$ is differentiable, and $\overline{\Omega}$ is not equal to $\mathbb{R}$. Let $U=\mathbb{R}-\overline{\Omega}$, and $]u,v[$ a connected component of $U$; we have $u,v\in \overline{\Omega}$.
Then for $x\in ]\frac{u+v}{2},v]$, we have $f(x)=v-x$, hence $f^{\prime}(x)=-1$. In particular, we have $f^{\prime}(v)=-1$. Then for $|x-v|$ small, $x\not =v$, we have $\displaystyle \frac{f(x)-f(v)}{x-v}\leq -\frac{1}{2}$. Hence as $f(v)=0$, for $x>v$, $x-v$ small, we get $f(x)<0$, a contradiction. The case of $]-\infty,v[$ and $]u,+\infty[$ can be shown to lead to a contradiction on the same way. So $\overline{\Omega}$ must be equal to $\mathbb{R}$