  Motivated by the need of a robust geometrical framework for the calculation
of long, and highly accurate waveforms for extreme-mass-ratio inspirals, this
work presents an extensive study of the hyperboloidal formalism for the Kerr
spacetime and the Teukolsky equation. In a first step, we introduce a generic
coordinate system foliating the Kerr spacetime into hypersurfaces of constant
time extending between the black-hole horizon and future null infinity, while
keeping track of the underlying degrees of freedom. Then, we express the
Teukolsky equation in terms of these generic coordinates with focus on
applications in both the time and frequency domains. Specifically, we derive a
wave-like equation in $2+1$ dimensions, whose unique solution follows directly
from the prescription of initial data (no external boundary conditions).
Moreover, we extend the hyperboloidal formulation into the frequency domain. A
comparison with the standard form of the Teukolsky equations allows us to
express the regularisation factors in terms of the hyperboloidal degrees of
freedom. In the second part, we discuss several hyperboloidal gauges for the
Kerr solution. Of particular importance, this paper introduces the minimal
gauge. The resulting expressions for the Kerr metric and underlying equations
are simple enough for eventual (semi)-analytical studies. Despite the
simplicity, the gauge has a very rich structure as it naturally leads to two
possible limits to extremality, namely the standard extremal Kerr spacetime and
its near-horizon geometry. When applied to the Teukolsky equation in the
frequency domain, we show that the minimal gauge actually provides the
spacetime counterpart of the well-known Leaver's formalism. Finally, we recast
the hyperboloidal gauges for the Kerr spacetime available in the literature
within the framework introduced here.
