This project explores quantum gravity models, focusing on the spectral properties of black holes, including quasinormal modes (QNMs), and the nature of gravitational singularities. Motivated by recent breakthroughs in astrophysics, such as the Event Horizon Telescope and LIGO detections, we investigate quantum modifications to general relativity within black hole physics, including metric perturbations and gravitational waves. Our objectives include developing and analyzing quantum models using noncommutative differential geometry to address gravitational perturbations of Schwarzschild, Kerr and slowly rotating Kerr black holes, revealing unique spectral features. We will study QNM spectra via semi-analytical and numerical methods, testing for the presence of isospectrality violations, and examine the potential ''smearing out'' of classical singularities through quantum mechanical probes. The methodological approach incorporates the algebra of infinitesimal diffeomorphisms and Hopf algebras to construct a deformed Leibniz rule. Furthermore, the project will employ an innovative framework based on noncommutative differential geometry, building on established methodologies in theoretical physics and fostering international collaboration.