Drell-Yan lepton pairs with finite transverse momentum are produced when the vector boson recoils against (multiple) parton emission(s), and is determined by QCD dynamics. At small transverse momentum, the fixed order predictions break down due to the emergence of large logarithmic contributions. This region can be studied via the $p_T^Z$ distribution constructed from the energies of the leptons, or through the $\phi^*$ distribution that relies on the directions of the leptons. For sufficiently small transverse momentum, the $\phi^*$ observable can be measured experimentally with better resolution. We study the small $p_T^Z$ and $\phi^*$ distributions up to next-to-next-to-leading order (NNLO) in perturbative QCD. We compute the $\phi^*$ distributions for the fully inclusive production of lepton pairs via $Z/\gamma^*$ to NNLO and normalise them to the NNLO cross sections for inclusive $Z/\gamma^*$ production. We compare our predictions with the $\phi^*$ distribution measured by the ATLAS collaboration during LHC operation at 8 TeV. We find that at moderate to large values of $\phi^*$, the NNLO effects are positive and lead to a substantial improvement in the theory-data comparison compared to next-to-leading order (NLO). At small values of $p_T^Z$ and $\phi^*$, the known large logarithmic enhancements emerge through and we identify the region where resummation is needed. We find an approximate relationship between the values of $p_T^Z$ and $\phi^*$ where the large logarithms emerge and find perturbative consistency between the two observables.