Let $M$ be an $n$-dimensional orientable non-compact manifold.
Is there an isomorphism as follows, and if so how can we construct it? (Or can you provide a reference?)
$$
H^{n-i}_{\operatorname{dR},c}(M, \mathbb R) \cong H_i(M,\mathbb R).
$$
On the left hand side we have de Rham cohomology with compact support and on the right hand side we have singular homology.
According to [1], Poincaré duality can be stated as saying that the integration pairing between compactly supported forms and forms induces an isomorphism:
$$
H^i_{\operatorname{dR}}(M,\mathbb R)\cong \left(H^{n-i}_{\operatorname{dR},c}(M, \mathbb R)\right)^\vee.
$$
Where $\bullet^\vee$ denotes the vector space dual. Now, note that $H^{n-i}_c\not\cong (H^i)^\vee$ in general, because in our situation the homology groups might be infinite dimensional. The de Rham theorem says that integration gives an isomorphism witht he dual of singular homology:
$$
H^i_{\operatorname{dR}}(M,\mathbb R)\cong \left(H_{i}(M, \mathbb R)\right)^\vee.
$$
Now, knowing that two (infinite dimensional) vector spaces have the same dual doesn't seem all that helpful. I don't even see how integration can give a map $H^{n-i}_c\to H_i$.
Another attempt would be to try to follow the proof that appears in [1] and [2]. You could try to "induct" by showing that if Poincaré duality holds in two open sets $U$ abd $V$ then it holds in the union. But the five-lemma together with the Mayer-Vietoris exact sequences aren't enough to construct a map, much less a canonical one.
I'd be happy with an answer that uses Verdier duality (bonus points for an answer with coefficients in a local system), but then the trouble is that books that talk about sheaves don't talk about homology (or they define it as the cohomology with compact support of the dual local system, in which case this is tautology). Then the question becomes:
For a local system $\mathcal L$ on $M$, are the following quasiisomorphic?
$$Rp_!(\mathcal L) \cong C_\bullet(M,\mathcal L):= C_\bullet(\widetilde M, \mathbb R)\otimes_{\mathbb R[\pi_1(M)]} \mathcal L_p.$$
Here $p:M\to *$ is the map to a point, $\widetilde M$ is the universal cover, and the fundamental group acts on $\widetilde M$ by deck transformations and on the stalk $\mathcal L_p$ by the monodromy of $\mathcal L$.
I guess this would be a corollary of the question
Is there a soft complex of sheaves that resolves $\mathcal L$, and whose compactly supported sections form a complex quasiisomorphic to $C_\bullet(M, \mathcal L)$?
Update: I've been looking at Glen Bredon's book [3], and Theorem V.9.2. is promising in that it relates homology and compactly supported cohomology. However, the definition of "sheaf homology" in that book doesn't seem to be related to singular homology as far as I can tell. In Chapter VI there is a relation between singular homology and Čech homology, but somehow not between these two and sheaf homology.
[1]: Greub, Werner; Halperin, Stephen; Vanstone, Ray, Connections, curvature, and cohomology. Vol. I: De Rham cohomology of manifolds and vector bundles, Pure and Applied Mathematics, 47. New York-London: Academic Press. XIX, 443 p. $ 31.00 (1972). ZBL0322.58001.
[2]: Hatcher, Allen, Algebraic topology, Cambridge: Cambridge University Press (ISBN 0-521-79540-0/pbk). xii, 544 p. (2002). ZBL1044.55001.
[3]: Bredon, Glen E., Sheaf theory., Graduate Texts in Mathematics. 170. New York, NY: Springer. xi, 502 p. (1997). ZBL0874.55001.
Best Answer
This is not a conclusive answer, but here's how I would approach this.
Let $\Delta_n(M)$ be the Abelian group of singular $n$-chains and $\Delta^n(M;\mathbb{R})=\mathrm{Hom}_\mathbb{Z}(\Delta_n(M),\mathbb{R})$ the $\mathbb R$-valued signular $n$-cochains. There are two subcomplexes that are relevant to the question: the complex $\Delta_*^\infty(M)$ of smooth singular chains (as explained in Bredon's book, for example), and the complex $\Delta_c^*(M)$ of compactly supported cochains, i.e. singular cochains that vanish on all chains with image outside of a compact set (which depends on the cochain).
Now, without having a reference or a proof, I would bet some money that the inclusion $\Delta_*^\infty(M)\hookrightarrow\Delta_*(M)$ is a chain homotopy equivalence. This should, in turn, dualize to chain homotopy equivalences $$\Delta^*(M;\mathbb R)\to\Delta^*_\infty(M;\mathbb R) \quad\text{and}\quad \Delta^*_c(M;\mathbb R)\to \Delta^*_{\infty,c}(M)$$ where $\Delta^n_\infty = \mathrm{Hom}(\Delta_n^\infty,\mathbb R)$ and $\Delta^n_{\infty,c}$ is the compactly supported analogue.
Next, integration gives rise to chain maps $$\Psi\colon \Omega^*(M)\to \Delta_\infty^*(M) \quad\text{and}\quad \Psi_c\colon \Omega^*_c(M)\to \Delta_{\infty,c}^*(M).$$ Bredon proves that $\Psi$ induces an isomorphism on cohomology and it should be possible to adapt the proof to show the same for $\Psi_c$.
Finally, the cohomology of $\Delta_c^*(M)$ is known a singular cohomology with compact supports, denoted by $H^*_c(M;\mathbb R)$. If $M$ has an orientation, then Poincaré duality gives isomorphisms $$H^{n-i}_c(M;\mathbb R)\cong H_{i}(M;\mathbb R)$$ with singular homology on the right hand side. And if everything above goes through as claimed, then the left hand side is isomorphic to compactly supported de Rham cohomology $H^{n-i}_{dR,c}(M)$.
I'll try to find some references later. Other duties are calling.