at.algebraic topology - Torsion in homology or fundamental group of subsets of Euclidean 3-space

I don't think that

torsion in the homology has been ruled out

Certainly, torsion in Cech cohomology has been ruled out for a compact subset. The "usual" universal coefficient formula, relating Cech cohomology to $operatorname{Hom}$ and $operatorname{Ext}$ of Steenrod homology, is not valid for arbitrary compact subsets of $Bbb R^3$ (although it is valid for ANRs, possibly non-compact). The "reversed" universal coefficient formula, relating Steenrod homology to $operatorname{Hom}$ and $operatorname{Ext}$ of Cech cohomology is valid for compact metric spaces, but it does not help, because $operatorname{Ext}(Bbb Z[frac1p],Bbb Z)simeqBbb Z_p/Bbb ZsupsetBbb Z_{(p)}/Bbb Z$, which contains $q$-torsion for all primes $qne p$. (Here $Bbb Z_{(p)}$ denotes the localization at the prime $p$, and $Bbb Z_p$ denotes the $p$-adic integers.
The two UCFs can be found in Bredon's Sheaf Theory, 2nd edition, equation (9) on p.292
in Section V.3 and Theorem V.12.8.)

The remark on $operatorname{Ext}$ can be made into an actual example. The $p$-adic solenoid $Sigma$ is a subset of $Bbb R^3$. The zeroth Steenrod homology $H_0(Sigma)$ is isomorphic by the Alexander duality to $H^2(Bbb R^3setminusSigma)$. This is a cohomology group of an open $3$-manifold contained in $Bbb R^3$, yet it is isomorphic to $Bbb Zoplus(Bbb Z_p/Bbb Z)$ (using the UCF, or the Milnor short exact sequence with $lim^1$), which contains torsion. Of course, every cocycle representing torsion is "vanishing", i.e. its restriction to each compact submanifold is null-cohomologous within that submanifold.

By similar arguments, $H_i(X)$ (Steenrod homology) contains no torsion for $i>0$ for every compact subset $X$ of $Bbb R^3$.

It is obvious that "Cech homology" contains no torsion (even for a noncompact subset $X$ of $Bbb R^3$), because it is the inverse limit of the homology groups of polyhedral neighborhoods of $X$ in $Bbb R^3$. But I don't think this is to be taken seriously, because "Cech homology" is not a homology theory (it does not satisfy the exact sequence of pair). The homology theory corresponding to Cech cohomology is Steenrod homology (which consists of "Cech homology" plus a $lim^1$-correction term). Some references for Steenrod homology are Steenrod's original paper in Ann. Math. (1940), Milnor's 1961 preprint (published in http://www.maths.ed.ac.uk/~aar/books/novikov1.pdf), Massey's book Homology and Cohomology Theory. An Approach Based on Alexander-Spanier Cochains, Bredon's book Sheaf Theory (as long as the sheaf is constant and has finitely generated stalks) and this paper http://front.math.ucdavis.edu/math/0812.1407

As for torsion in singular $4$-homology of the Barratt-Milnor example, this is really a question about framed surface links in $S^4$ (see the proof of theorem 1.1 in the linked paper).