When thinking about multipole moments in everyday life, the first thing that came to my mind was the magnetic dipole moment of earth which is (or at least used to be) very helpful for not getting lost at sea.
After some thought, it very quickly became clear to me that the earth’s magnetic moment is only a dipole in approximation. This is the case because the earth is not a perfect system, which means that the convection currents of molten metal that create the magnetic moment are irregular and so is the resulting magnetic moment.
In general, it seems to me that in order to observe a “true” dipole, one has to consider smaller and simpler systems such as the magnetic dipole moments of electrons.