  One of the main predictions of general relativity is the existence of black
holes featuring a horizon beyond which nothing can escape. Gravitational waves
from the remnants of compact binary coalescences have the potential to probe
new physics close to the black hole horizons. This prospect is of particular
interest given several quantum-gravity models that predict the presence of
horizonless and singularity-free compact objects. The membrane paradigm is a
generic framework that allows one to parametrise the interior of compact
objects in terms of the properties of a fictitious fluid located at the
object's radius. It has been used to derive the quasinormal mode spectrum of
static horizonless compact objects. Extending the membrane paradigm to rotating
objects is crucial to constrain the properties of the spinning merger remnants.
In this work, we extend the membrane paradigm to linear order in spin and use
it to analyse the relationships between the quasi-normal modes, the object's
reflectivity, and the membrane parameters. We find a breaking of isospectrality
between axial and polar modes when the object is partially reflecting or the
compactness differs from the black hole case. We also find that in reflective
ultracompact objects some of the modes tend towards instability as the spin
increases. Finally, we show that the spin enhances the deviations from the
black-hole quasinormal mode spectrum as the compactness decreases. This implies
that spinning horizonless compact objects may be more easily differentiated
than nonspinning ones in the prompt ringdown.
