Edit: I have since resolved my question.
If X is locally compact Hausdorff in addition to being cohomologically locally contractible with coefficients in $A$ - eg it is a manifold or an open subset of some locally finite CW complex - then, functorially in proper continuous maps between such spaces, compactly supported singular and sheaf cohomology coincide with coefficients in the Abelian group $A$. It requires only a minor addition to Sella’s argument (you can use the same flasque resolution they use), but I don’t have time to write it up right now.
I'm a little way into reading this nice paper by Yehonatan Sella, which claims - after minor generalisations I was able to see - the following:
Let the Abelian group $A$ be fixed. Let "clc" be the following property of topological spaces $X$: $$\forall x\in X,\,\forall n\in\Bbb N_0:\quad\varinjlim_{\quad\,\,U\ni x\\U\subseteq X\text{ is open }}\mathrm{H}^n_{\mathrm{sing}}(U,x;A)=0$$
There is a functorial isomorphism $\mathrm{H}^\bullet_{\mathrm{sing}}(-;A)\cong\mathscr{H}^\bullet(-;\underline{A})$ on the full subcategory of clc spaces, where $\underline{A}$ denotes the 'constant' sheaf functor.
The isomorphism is also natural in the group $A$. However, I'm quite fond of Iversen's book "Cohomology of Sheaves"; therein, compactly supported cohomology comes up often. I wonder to what extent we can have $\mathrm{H}^\bullet_{\mathrm{sing},C}(X;A)\cong\mathscr{H}^\bullet_C(X;\underline{A})$. It is certainly true when $X$ is a smooth manifold. I expect a more general answer to this is known, but I haven't found anything.
The proof in Sella's paper goes as follows: a "functor" from spaces to complexes of sheaves is constructed, with the resulting $\mathscr{C}^\bullet_{X,A}$ being exact in positive degrees and having zeroth cohomology the sheaf $\underline{A}$. We prove the $\mathscr{C}^n_{X,A}$ are flabby/flasque sheaves, also, so we realise $\mathscr{H}^n(X;\underline{A})\cong\mathrm{H}^n\mathscr{C}^\bullet_{X,A}(X)$. However, in the construction there is a (surjective) quasiisomorphism $\mathrm{C}^\bullet_{\mathrm{sing}}(X;A)\to\mathscr{C}^\bullet_{X,A}(X)$, so we get the result (for individual $X$). Naturality in $X$ is not so hard to conclude either; if $f:X\to Y$ is a map of clc spaces there is a sheaf morphism $\mathscr{C}^\bullet_{Y,A}\to f_\ast\mathscr{C}^\bullet_{X,A}$ which satisfies the requirements of Scholium $2.5.2$ in Iversen and we realise this morphism computes $f^\ast$ on sheaf cohomology, but also that it is compatible with the usual restriction map $\mathrm{C}^\bullet_{\mathrm{sing}}(Y;A)\to\mathrm{C}^\bullet_{\mathrm{sing}}(X;A)$.
$\mathscr{C}^\bullet_{X,A}(U):=\varinjlim_{\eta\text{ is a nesting on }U}C^\bullet_\eta(U;A)$, the colimit being taken in Abelian groups, where $C^k_\eta$ and "nesting" are defined in the paper. Using naturality I'm able to conclude the following thing:
If $X$ is a Hausdorff clc space, $\mathrm{H}^\bullet_{\mathrm{sing},C}(X;A)\cong\mathrm{H}^\bullet(\varinjlim_{K\subseteq X\text{ compact }}\ker(\mathscr{C}_{X,A}^\bullet(X)\to\mathscr{C}^\bullet_{X,A}(X\setminus K)))$
But it is very unclear to me if that object in the parantheses can be construed as $\Gamma_C^\bullet$ of any kind of soft resolution of $\underline{A}$. We would have to assume $X$ to be LCH for that to work, too. I once saw it said that this is true for 'reasonable' spaces, but no reference or elaboration was given.
Does anyone here have a reference or elaboration? It is known if there is a (hopefully functorial) isomorphism between the two cohomologies for a broader class of space?
