We study the regularity properties of Gaussian fields defined over spheres across time. In particular, we consider two alternative spectral decompositions for a Gaussian field on S 2 × R. For each decomposition, we establish regularity properties through Sobolev and interpolation spaces. Then, conditions for sample Hölder continuity are also provided. We propose a simulation method and study its level of accuracy in the L 2 sense. The method turns to be both fast and efficient.