A large-scale spanwise and wall-normal array of sonic anemometers in the atmospheric surface layer is used to acquire all three components of instantaneous fluctuating velocity as well as temperature in a range of stability conditions. These data permit investigation of the three-dimensional statistical structure of turbulence structures. Based on a similar dataset, Krug et al. (Boundary-Layer Meteorol., vol. 172, 2019, pp. 199-214) reported a self-similar range of wall-attached turbulence structures under both unstable and near-neutral stability conditions. They considered only a wall-normal array and thus assessed statistical structure in the wall-normal direction, in relation to the streamwise wavelength. The present work extends the view of a self-similar range of turbulence structures, by including the statistical structure in the spanwise direction. Moreover, by analysing the phase shift between synchronized measurements in the spectral domain, it is inferred how a scale-dependent inclination angle in the streamwise/wall-normal plane varies with stability. Results suggest that the self-similar wall-attached structures have similar aspect ratios between streamwise/wall-normal scales and streamwise/spanwise scales such that for both near-neutral and unstable conditions. Under the most unstable conditions, coherent structures with are inclined at angles as high as relative to the solid boundary, while larger scales exhibit inclination angles of approximately. For near-neutral stability conditions, the angle tends towards for all scales. It is noted that in the near-neutral condition, the structure inclination angle and the aspect ratio - and thus the statistical modelling of coherent structures in the atmospheric surface layer - are highly sensitive to the value of the stability parameter.