Is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Q}/\mathbb{Z})\cong\bigoplus_p\mathbb{Q}_p$? Is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Q}/\mathbb{Z})\cong\bigoplus_p\mathbb{Q}_p$? Or maybe $\prod_p\mathbb{Q}_p$?
I know $\mathbb{Q}/\mathbb{Z}\cong\bigoplus_p \mathbb{Z}_{p^\infty}$, and also that $\operatorname{Hom}(\mathbb{Q},\mathbb{Z}_{p^\infty})\cong\mathbb{Q}_p$. So I want to say
$$
\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Q}/\mathbb{Z})\cong\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\bigoplus_p\mathbb{Z}_{p^\infty})\cong\bigoplus_p\operatorname{hom}(\mathbb{Q},\mathbb{Z}_{p^\infty})\cong\bigoplus_p\mathbb{Q}_p
$$
However, I'm not sure about the middle isomorphism. I only know of rules which allow one to pull a coproduct in the first term out into the front of Hom and change it to a product, or you can pull a product in the second term out into the front of Hom as a product. 
Some digging here seems to imply that one cannot generally pull a direct sum in the second term out in front as a product, ans this is an isomorphism if the first term is finitely generated, but $\mathbb{Q}$ is certainly not a f.g. $\mathbb{Z}$-module.
 A: From this answer, we know that
$$\hom(\mathbb{Q}/\mathbb{Z}, \mathbb{Q}/\mathbb{Z}) \cong \hat{\mathbb{Z}} \cong \prod_p \mathbb{Z}_p$$
We can therefore compute the long exact sequence
$$ 0 \to \hom(\mathbb{Q}/\mathbb{Z}, \mathbb{Q}/\mathbb{Z})
\to \hom(\mathbb{Q}, \mathbb{Q}/\mathbb{Z})
\to \hom(\mathbb{Z}, \mathbb{Q}/\mathbb{Z})
\to \operatorname{Ext}^1(\mathbb{Q}/\mathbb{Z}, \mathbb{Q}/\mathbb{Z})
\to \ldots
$$
to be the short exact sequence
$$ 0 \to \hat{\mathbb{Z}} \to \hom(\mathbb{Q}, \mathbb{Q}/\mathbb{Z})
\to \mathbb{Q} / \mathbb{Z} \to 0$$
The first map is given by
$$x \mapsto \left(\frac{m}{n} \mapsto \frac{x'm}{n} \right)
\qquad \qquad (x' \equiv x \bmod n)$$ 
and the second is
$$ f \mapsto f(1) $$
Furthermore, $\hom(\mathbb{Q}, \mathbb{Q}/\mathbb{Z})$ is a $\mathbb{Q}$-vector space; the action of $q$ on the function $f$ gives the function $(q \cdot f)(x) = f(qx)$, so the tensor product with $\mathbb{Q}$ gives an exact sequence
$$ 0 \to \mathbb{Q} \otimes \hat{\mathbb{Z}} \to \mathbb{Q} \otimes \hom(\mathbb{Q}, \mathbb{Q}/\mathbb{Z}) \to \mathbb{Q} \otimes \mathbb{Q}/\mathbb{Z} \to 0$$
which simplifies to
$$ \mathbb{Q} \otimes \hat{\mathbb{Z}} \cong \hom(\mathbb{Q}, \mathbb{Q}/\mathbb{Z})$$
A: Not quite. The Pontryagin dual of $\mathbf Q$ is a weird object, called the solenoid. Since $\mathbf Q = \varinjlim \frac{1}{n}\mathbf Z$, where the limit is taken over the integers ordered by divisibility, it follows that
$$\widehat{\mathbf Q/\mathbf Z} = \varprojlim \widehat{\frac{1}{n}\mathbf Z}$$
We can replace $\frac{1}{n}\mathbf Z$ by $\mathbf Z$ in the limit, if we also replace the morphisms in the limit with the appropriate multiplication maps. From Fourier theory, the dual of $\mathbf Z$ is the circle group $S^1$, and therefore
$$\widehat{\mathbf Q/\mathbf Z} = \varprojlim S_n,$$
where $S_n=S^1$ for each $n$ and for any $n,m$, the map $S_{nm} \to S_m$ is multiplication by $n$.
(A word of warning: Here, I am using the "full" Pontryagin dual, namely the maps into $S^1$, rather than into $\mathbf Q/\mathbf Z$. Otherwise, the dual of $\mathbf Z$ is not $S^1$ but the torsion $\mathbf Q/\mathbf Z\subseteq S^1$. The solenoid is actually bigger than the group you asked about: your group consists of the topologically nilpotent elements of the solenoid, i.e. those $x$ such that $x^{n!}$ converges to $1$. Remark however that the solenoid has no torsion elements.)
The solenoid is a bizarre topological space, one of the simplest examples of an indecomposable continuum.
Objects like the solenoid appear naturally in the fourier theory of number fields. If $\mathbf A$ denotes the adele ring of $\mathbf Q$, then one has an exact sequence
$$0 \to \mathbf Q \to \mathbf A \to S^1 \to 1$$
where the last map is given by the adelic exponential. Taking duals, and using the self-duality of adeles, we get a corresponding exact sequence 
$$0 \to \mathbf Z \to \mathbf A \to \widehat{\mathbf Q} \to 1$$
Thus, the group of additive adeles appears naturally as a (non-split) extension of the solenoid by $\mathbf Z$. 
Interestingly, the solenoid admits an embedding as a compact subspace of $\mathbf R^3$.
