Is the Hurewicz theorem ever used to compute abelianizations? The Hurewicz theorem tells us that if $X$ is a path-connected space then $H_1(X, \, \mathbb{Z})$ is isomorphic to the abelianisation of $\pi_1(X)$.  This gives a potential method for computing the abelianisation of a (sufficiently nice) group $G$: realise it as the fundamental group of a space $X$ and then compute $H_1(X, \, \mathbb{Z})$ by your favourite means.  

Is this method ever used in practice?  Are there nice examples of abelianisations which are easily (best?) computed in this way?

 A: Yes, the fundamental group of the Hawaiian earring $\pi_1(\mathbb{H},b_0)$ is an important group which is sometimes called the free sigma product $\#_{\mathbb{N}}\mathbb{Z}$. Its is often defined in purely algebraic terms as a group of "transfinite words" in countably many letters. In many ways this group behaves like the non-abelian version of the Specker group $\prod_{\mathbb{N}}\mathbb{Z}$ and it is the key to the homotopy classification of 1-dimensional Peano continua. However, it's abelianization is not $\prod_{\mathbb{N}}\mathbb{Z}$; it is a good deal more complicated. The abelianization was first described by Eda and Kawamura in the following paper:
The Singular Homology Group of the Hawaiian Earring, Journal of the London Mathematical Society, 62 no. 1 (2000) 305–310.
The proof explicitly uses the Hawaiian earring and the authors mention in Remark 2.7 that 

Though the question itself is formulated algebraically, we have not succeeded in finding a purely algebraic proof of it.

One way to represent the isomorphism class of this abelianization is as $$Ab(\#_{\mathbb{N}}\mathbb{Z})\cong \prod_{\mathbb{N}}\mathbb{Z}\oplus \prod_{\mathbb{N}}\mathbb{Z}/\bigoplus_{\mathbb{N}}\mathbb{Z}$$
Similar methods have been used to identify abelienizations of fundamental groups of other 1-dimensional spaces, e.g. the Menger curve.
A: A usually hard problem in Algebraic Geometry is computing the fundamental group of the complement $U=\mathbb{P}^n - V$, where $V$ is a reduced hypersurface. 
At least in principle, the computation of $\pi_1(U)$ can be carried out by using Seifert-Van Kampen theorem; the tricky part is that the relations in the presentation depend not only on the singularities of $V$, but also on their mutual position. 
A classical example due to Zariski is when $V \subset \mathbb{P}^2$ is a curve of degree $6$ having six ordinary cusps and no other singularities. Then $\pi_1(U)$ is the free product $\mathbb{Z}/2 \mathbb{Z} * \mathbb{Z}/3 \mathbb{Z}$ if the six cusps lie on the same conic, and the direct product $\mathbb{Z}/ 2 \mathbb{Z} \times \mathbb{Z}/3 \mathbb{Z}$ otherwise.   
However, by using Lefschetz duality we can compute in a straightforward way $H_1(U, \, \mathbb{Z})$ for all hypersurfaces, so that the abelianization of $\pi_1(U)$ is always known even if a presentation for $\pi_1(U)$ is not available. In fact, there is the following result that can be found in A. Dimca's book Singularities and topology of hypersurfaces, Chapter 4.

Proposition. Assume that the hypersurface $V \subset \mathbb{P}^n$ has $k$ irreducible components $V_1, \ldots, V_k$ of degrees $d_1, \ldots, d_k$. Let $d$ be the greatest common divisor of the integers $d_1, \ldots, d_k$. Then $$H_1(U, \, \mathbb{Z}) = \mathbb{Z}^{k-1} \oplus \mathbb{Z}/ d \mathbb{Z}.$$  

In particular, $H_1(U)$ does not depend on the singularities of $V$, but only on the number and degrees of its irreducible components. As a consequence, we recover the fact that in both cases of Zariski's example quoted above (where $k=1, \, d_1=6$) the abelianization of $\pi_1(U)$ is the cyclic group of order $6$.
A: The mapping class group of a smooth manifold $M$ is the group of all its self diffeomorphisms up to isotopy, i.e. $\pi_0(\operatorname{Diff}(M))\cong \pi_1(B\operatorname{Diff}(M))$.
A large portion of geometric topology is concerned with gaining a better understanding of the homotopy type of $B\operatorname{Diff}(M)$, in particular of its cohomology as the latter is the ring of characteristic class of smooth $M$ bundles.
In the last 15 years, initiated by Madsen and Weiss' solution of the Mumford conjecture, the understanding of the homology of $B\operatorname{Diff}(M)$ has fundamentally improved which led, among others, to calculations of $H_1(B\operatorname{Diff}(M))\cong \pi_0(\operatorname{Diff}(M))^{ab}$ for some $M$, e.g. by Galatius and Randal-Williams in the following paper.
Abelian quotients of mapping class groups of highly connected manifolds, Mathematische Annalen
June 2016, Volume 365, Issue 1, pp 857–879
