  We study the stability of quasinormal modes (QNM) in asymptotically flat
black hole spacetimes by means of a pseudospectrum analysis. The construction
of the Schwarzschild QNM pseudospectrum reveals the following: (i) the
stability of the slowest-decaying QNM under perturbations respecting the
asymptotic structure, reassessing the instability of the fundamental QNM
discussed by Nollert [H. P. Nollert, About the Significance of Quasinormal
Modes of Black Holes, Phys. Rev. D 53, 4397 (1996)] as an "infrared" effect;
(ii) the instability of all overtones under small-scale ("ultraviolet")
perturbations of sufficiently high frequency, which migrate towards universal
QNM branches along pseudospectra boundaries, shedding light on Nollert's
pioneer work and Nollert and Price's analysis [H. P. Nollert and R. H. Price,
Quantifying Excitations of Quasinormal Mode Systems, J. Math. Phys. (N.Y.) 40,
980 (1999)]. Methodologically, a compactified hyperboloidal approach to QNMs is
adopted to cast QNMs in terms of the spectral problem of a non-self-adjoint
operator. In this setting, spectral (in)stability is naturally addressed
through the pseudospectrum notion that we construct numerically via Chebyshev
spectral methods and foster in gravitational physics. After illustrating the
approach with the P\"oschl-Teller potential, we address the Schwarzschild black
hole case, where QNM (in)stabilities are physically relevant in the context of
black hole spectroscopy in gravitational-wave physics and, conceivably, as
probes into fundamental high-frequency spacetime fluctuations at the Planck
scale.
