The stable surface boundary layers observed in the experiments SABLES and BORRIS95 have been investigated by means of detailed profiles of high resolution temperature and wind profile measurement together with turbulence measurements at several levels. The third order structure functions of horizontal velocity and temperature have been investigated ruling out the presence of 2D turbulence. The profiles of temperature show that the boundary layer separates into a shallow layer (5 to 15 m) near the ground where the Monin-Obukhov profiles apply and a stable layer aloft where M-O scaling from the lower layer does not apply. Analysis shows that the separation of the lower layer and the layer aloft is associated with the presence of supercritical Richardson numbers. The low frequency behaviour of the spectra of temperature and wind speed show slopes of k^(-2) for medium wind speeds whereas low wind speeds cause a k^(-3) slope. The slope of k^(-2) indicates a situation where shear instabilities arise causing oscillations in both temperature and velocity and is associated with a persistent oscillation in the wind direction. For very low wind speed the slopes indicate the presence of waves.