Black hole spectroscopy is a powerful tool to probe the Kerr nature of astrophysical compact objects and their environment. The observation of multiple ringdown modes in gravitational waveforms could soon lead to high-precision gravitational-wave spectroscopy, thus it is critical to understand if the quasinormal mode spectrum itself is affected by astrophysical environments, quantum corrections, and other generic modifications. In this talk, I will review the black-hole spectroscopy program and its challenges regarding quasinormal mode detection, the overtone status and the recent evidence that supports the existence of nonlinearities in the spectrum of black holes. I will then discuss a newly introduced non-modal tool in black physics, namely the pseudospectrum; a mathematical notion which can shed light on the spectral stability of quasinormal modes, and discuss its novel applications in black holes and exotic compact objects. I will show that quasinormal modes generically suffer from spectral instabilities and argue how such phenomenon can further affect black hole spectroscopy.