An interesting application (in my opinion) of multipoles is found in General Relativity (GR). The mass quadrupole moment can (if it changes in time) cause gravitational waves (higher order ones can also but their effects are way smaller). The mass monopole moment is simply the total mass (which is conserved) and the dipole moment is related to total momentum (also conserved), so these cannot change in time and cause gravitational waves. These gravitational waves can for example be detected during the merger of two black holes (or neutron stars).