  Accretion disks are highly unstable to magnetic instabilities driven by shear
flow, where classically, the axisymmetric, weak-field Magneto-Rotational
Instability (MRI) has received much attention through local WKB approximations.
In contrast, discrete non-axisymmetric counterparts require a more involved
analysis through a full global approach to deal with the influence of the
nearby magnetohydrodynamic (MHD) continua. Recently, rigorous MHD spectroscopy
identified a new type of an ultra-localised, non-axisymmetric instability in
global disks with super-Alfv\'enic flow. These Super-Alfv\'enic Rotational
Instabilities (SARIs) fill vast unstable regions in the complex eigenfrequency
plane with (near-eigen)modes that corotate at the local Doppler velocity and
are radially localised between Alfv\'enic resonances. Unlike discrete modes,
they are utterly insensitive to the radial disk boundaries. In this work, we
independently confirm the existence of these unprecedented modes using our
novel spectral MHD code Legolas reproducing and extending our earlier study
with detailed eigenspectra and eigenfunctions. We calculate growth rates of
SARIs and MRI in a variety of disk equilibria, highlighting the impact of field
strength and orientation, and find correspondence with analytical predictions
for thin, weakly magnetised disks. We show that non-axisymmetric modes can
significantly extend instability regimes at high mode numbers, with maximal
growth rates comparable to the MRI. Furthermore, we explicitly show a region
filled with quasi-modes whose eigenfunctions are extremely localised in all
directions. These modes must be ubiquitous in accretion disks, and play a role
in local shearing box simulations. Finally, we revisit recent dispersion
relations in the Appendix, highlighting their relation to our global framework.
