I realized perhaps my question is equivalent to asking for a characterization of sets $S\subseteq [a,b]$ that can not support a singular continuous measure.

Actually, here we may assume that $S$ is the set of all points of non-differentiability of $F$, so that $S$ is Borel.

It "is well-known that assuming the countable axiom of choice that every uncountable Borel set contains a [nonempty] perfect set."

So, if $S$ is uncountable, then there is an injective continuous (and hence strictly monotonic) function $g\colon C\to S$, where $C$ is the Cantor set (see e.g. Theorem 6.2), which implies that $S$ can support a singular continuous measure.

Thus (assuming the countable axiom of choice), the set $S$ of all points of non-differentiability of $F$ cannot support a singular continuous measure if and only if $S$ is countable.

I realized perhaps my question is equivalent to asking for a characterization of sets $S\subseteq [a,b]$ that can not support a singular continuous measure.

Actually, here we may assume that $S$ is the set of all points of non-differentiability of $F$, so that $S$ is Borel.

So, by Theorem 13.6, if $S$ is uncountable, then $S$ contains a homeomorphic image of the Cantor set, which implies that $S$ can support a singular continuous measure.

Thus (assuming the countable axiom of choice), the set $S$ of all points of non-differentiability of $F$ cannot support a singular continuous measure if and only if $S$ is countable.