In the case of $\Bbb Z$ coefficients, you seem to need exactly the same hypotheses as you would for $\Bbb R$ coefficients, or any Abelian group for that matter.
Namely, the hypothesis which can be used for Sella’s paper or Petersen’s paper on this - for proving our result with coefficients in the Abelian group $G$:
$$\forall x\in X,\,\forall n\in\Bbb N_0\,:\\\varinjlim_{U\text{ an open neighbourhood of }x}\mathrm{H}_{\mathrm{sing}}^n(U,x;G)=0$$
“Cohomologically locally contractible”. Then $\mathrm{H}^\bullet_{\mathrm{sing}}(X;G)\cong\mathcal{H}^\bullet(X;\underline{G})$ (functorially in spaces satisfying that hypothesis).
About the De Rham theorem: if you want an isomorphism of (compactly supported or vanilla) cohomology groups of a manifold, or for compactly supported De Rham cohomology with compactly supported sheaf cohomology, see Iversen’s “cohomology of sheaves”. You can argue a certain De Rham complex of sheaves is a soft resolution of the constant $\Bbb R$ sheaf, and for manifolds that’s good enough to compute compactly supported (or globally supported) sheaf cohomology.
This gives a De Rham comparison with singular and De Rham cohomology groups until you couple the result with the result of Sella/Petersen. I would question your usage of “at once”, since either paper is fairly technical and machinery heavy :) If you want the much stronger theorem of an isomorphism of cohomology rings… I’m not currently convinced if the quoted result is good enough for that. Sella’s paper does not even argue the isomorphism is functorial in the space; I was able to make this extension, but I haven’t yet managed to figure out if we can adapt their technique to get cup product results. You may have to use the standard De Rham integration method… sheaf theoretic techniques can prove the global De Rham and sheaf cohomologies of a manifold are isomorphic as rings, but I’m not sure in what generality sheaf cohomology and singular cohomology are isomorphic as rings. There is also the trouble that different authors seem to have used apparently different definitions of the cup product in sheaf cohomology.
Edit: I have since figured it out. Sella’s method yields a cup product isomorphism with no extra hypotheses on the space. I have no idea if this is proven anywhere (other than my own head), but I intend to write it up at some point. It essentially follows from the final theorem of Iversen in the section where they discuss using localisations at pointwise homotopy equivalences. I don’t think their acyclic resolution is such an equivalence but in fact that does not matter.
