We present a microscopic analysis of the collective behaviour of the lead isotopes in the vicinity of $^{208}$Pb. In this study, we rely on a coherent approach based on the Generator Coordinate Method including exact projection on $N$ and $Z$ numbers within a collective space generated by means of the constrained Hartree-Fock BCS method. With the same Hamiltonian used in HF+BCS calculations, we have performed a comprehensive study including monopole, quadrupole and octupole excitations as well as pairing vibrations. We find that, for the considered nuclei, the collective modes which modify the most the conclusions drawn from the mean-field theory are the octupole and pairing vibrations.