The fine structure and dynamics of magnetic domain walls in ultrathin films with perpendicular magnetization, in presence of a secondary anisotropy, is analysed owing to micro-magnetics. Two cases are considered, a cubic anisotropy typical for (111) oriented garnet epitaxial films, and an orthorhombic anisotropy as found in e.g. Co/W(110) films. The statics is solved first, showing that, in general, domain walls are not of the pure Bloch type. The dynamics under the spin Hall effect induced by a current flowing in an adjacent layer is then monitored. Finite and non-negligible domain wall velocities are predicted in both cases, in the absence of Dzyaloshinskii-Moriya interactions, with distinct behaviours regarding the current density and its orientation with respect to the secondary anisotropy axes. The relevance of these results to recent reports of current driven domain wall dynamics in insulating ultrathin garnet films, capped with platinum, is discussed.