  We define and study natural $\mathrm{SU}(2)$-structures, in the sense of
Conti-Salamon, on the total space $\cal S$ of the tangent sphere bundle of any
given oriented Riemannian 3-manifold $M$. We recur to a fundamental exterior
differential system of Riemannian geometry. Essentially, two types of
structures arise: the contact-hypo and the non-contact and, for each, we study
the conditions for being hypo, nearly-hypo or double-hypo. We discover new
double-hypo structures on $S^3\times S^2$, of which the well-known
Sasaki-Einstein are a particular case. Hyperbolic geometry examples also
appear. In the search of the associated metrics, we find a theorem, useful for
explicitly determining the metric, which applies to all
$\mathrm{SU}(2)$-structures in general. Within our application to tangent
sphere bundles, we discover a whole new class of metrics specific to
3d-geometry. The evolution equations of Conti-Salamon are considered; leading
to a new integrable $\mathrm{SU}(3)$-structure on ${\cal S}\times\mathbb{R}_+$
associated to any flat $M$.
