I am not sure about cohomology with coefficients in $\mathbb Z$, but if we replace $\mathbb Z$ by $\mathbb Q$ and if $\Omega$ is a ball, then Poincar\'e duality implies that $H^1(\Omega\setminus S,\mathbb Q)$ is dual to $H^{2n-1}_c(\Omega\setminus S,\mathbb Q)$ (cohomology with compact support). Now the exact sequence
$$
H^{2n-2}_c(\Omega)\to H^{2n-2}_c(S)\to H^{2n-1}_c(\Omega\setminus S)\to H^{2n-1}_c(\Omega)
$$
shows that $H^{2n-1}_c(\Omega\setminus S)\cong H^{2n-2}_c(S)$; the latter is isomorphic to $H^{2n-2}_c(S_{\mathrm{smooth}})$, that is, to $\mathbb Q^N$, where $N$ is the number of components of $S$.