  We initiate the study of dynamical instabilities of higher-dimensional black
holes using the blackfold approach, focusing on asymptotically flat boosted
black strings and singly-spinning black rings in $D\ge5$. We derive novel
analytic expressions for the growth rate of the Gregory-Laflamme instability
for boosted black strings and its onset for arbitrary boost parameter. In the
case of black rings, we study their stability properties in the region of
parameter space that has so far remained inaccessible to numerical approaches.
In particular, we show that very thin (ultraspinning) black rings exhibit a
Gregory-Laflamme instability, giving strong evidence that black rings are
unstable in the entire range of parameter space. For very thin rings, we show
that the growth rate of the instability increases with increasing
non-axisymmetric mode $m$ while for thicker rings, there is competition between
the different modes. However, up to second order in the blackfold
approximation, we do not observe an elastic instability, in particular for
large modes $m\gg1$, where this approximation has higher accuracy. This
suggests that the Gregory-Laflamme instability is the dominant instability for
very thin black rings. Additionally, we find a long-lived mode that describes a
wiggly time-dependent deformation of a black ring. We comment on disagreements
between our results and corresponding ones obtained from a large $D$ analysis
of black ring instabilities.
