There is a geometric version of the definition of singular homology $H_n$ in terms of continuous maps of "pseudomanifolds" ($n$-dimensional simplicial complexes such that every $(n-1)$-simplex is contained in exactly two $n$-simplices with opposite induced orientations and every simplex is contained in some $n$-simplex) and a notion of cobordism between such maps. Maps from smooth manifolds are a special case of maps from pseudomanifolds, so there is induced a natural map
$MSO_k X \longrightarrow H_k X.$
(This question phrases it better: Is there a initial "bordism-like" homology theory?)
These theories are of representable, in the sense that
$MSO_k X = \pi_k(MSO \wedge X)$ and $H_k X = \pi_k(H\mathbb{Z} \wedge X)$.
Is there a map $MSO \to H\mathbb Z$ of spectra inducing this natural homomorphism? If so, what can one say about it?
I guess I should actually write up an answer instead of posting comments.
We seek a map $MSO \to H\mathbb{Z}$. It's not hard to see that any such map factors through $MSO \to H\pi_0 MSO$, and homotopy classes of maps from $H\pi_0 MSO \to H\mathbb{Z}$ are in bijection with $\pi_0 MSO \to \mathbb{Z}$, so we just need to understand the map $MSO \to H\mathbb{Z}$ in $\pi_0$.
Now we can think of $\pi_0 MSO$ as $0$-dimensional manifolds up to cobordism, and we can think of $\pi_0 H\mathbb{Z} \cong \mathbb{Z}$ as $0$-dimensional "pseudomanifolds" up to cobordism. But $0$-manifolds are the same as $0$-pseudomanifolds, so the map $\pi_0 MSO \cong \mathbb{Z} \to \mathbb{Z}$ is the identity.
This shows that the map $H\pi_0 MSO \xrightarrow{\sim} H\mathbb{Z}$ is an equivalence, so the map $MSO \to H\mathbb{Z}$ is just the zeroth Postnikov stage $$MSO \to H\pi_0 MSO \simeq H\mathbb{Z}.$$