  We investigate the effect of superradiant scattering of gravitational
perturbations on the stability of rotating black strings, focusing on the six
dimensional equal-spinning Myers-Perry black string. We find that rapidly
rotating black strings are unstable to gravitational superradiant modes within
a bounded range of string lengths. The instability occurs because momentum
along the string direction creates a potential barrier that allows for the
confinement of superradiant modes. Yet, five dimensional Myers-Perry black
holes do not have stable particle orbits so, unlike other known superradiant
systems, these black strings remain stable to perturbations with sufficiently
high azimuthal mode number -- this is a `finite-$m$' superradiant instability.
For some parameters, this instability competes with the Gregory-Laflamme
instability, but otherwise exists independently. The onset of this instability
is degenerate and branches to multiple steady-state solutions. This paper is
the first of a trilogy: in the next two, we construct two distinct families of
rotating strings emerging from the superradiant onset (the `black resonator
strings' and `helical black strings'). We argue that similar physics is present
in 5-dimensional Kerr black strings, but not in $D>6$ equal-spinning
Myers-Perry black strings.
