One of the most fundamental and classical problems in Fourier analysis is the pointwise almost-everywhere convergence of partial Fourier integrals, a problem that remains largely open in higher dimensions. This naturally leads to the closely related problem of almost-everywhere localisation: if a function is supported in a region $\Omega$, do its partial Fourier integrals converge to zero almost everywhere away from $\Omega$? One approach to this problem follows the philosophy of Sj"olin, which connects localisation to the local regularity of spherical means in the radial variable. In this talk, we first discuss the localisation problem and the local regularity of spherical means in the Euclidean setting, and then investigate these questions in the setting of harmonic NA groups. This talk is based on joint work with Swagato K. Ray.