Not so much in terms of high dimensional probability (though this is definitely a high dimensional probability question). 
I suspect however that you meant a slightly different question, which I answer below.

If $S$ is a subset of $R$ and $X$ is say a Gaussian process then the [Kac-Rice formula](https://en.wikipedia.org/wiki/Rice%27s_formula)
allows you to compute $E\{\#s: X_s=0\})$ (the number with $\neq 0$ will be typically infinity if the marginal has a density). Dito if $S$ is a subset of $R^k$ and $X$ is a $k$-dimensional vector. Kac-Rice is not limited to the Gaussian setup, but the latter simplifies the computation.

Note that in the one dimensional case, if $X$ is very irregular (e.g., Brownian motion) then the expectation above is $\infty$.

If you really meant what you asked, then in case $P(X_s=0)<1$ and $S$ is one dimensional of positive Lebesgue measure, the answer is $\infty$ by Fubini.