Abstract. We study four intervals of Cluster data, lasting from five to eight hours, in the flanks of the magnetosheath. In a first part, we make numerical simulations of these magnetosheath crossings, using a three-dimensional double-adiabatic MHD model of the magnetosheath and assuming that the proton temperature anisotropy is bounded by the kinetic thresholds of the Alfvén proton cyclotron instability and of the mirror instability. The conditions at the upstream boundary of the numerical domain are given by the solar wind parameters observed by ACE. We assume that the magnetopause is a fixed and impenetrable boundary, i.e. without magnetic reconnection. The global agreement between the observations and the simulations confirms the validity of the model in the magnetosheath flanks. We discuss the consequences of different models of the magnetopause on some simulation results. In a second part, we compare the observed proton temperature anisotropy and the kinetic anisotropy thresholds of the two above-mentioned instabilities which are local functions of the proton β. In the intervals with a low proton β, the observed temperature anisotropy agrees well with the kinetic threshold of the proton-cyclotron instability; in the intervals with a higher β, the observed anisotropy is close to both the proton-cyclotron and the mirror thresholds. This confirms that the observed proton anisotropy is indeed bounded by the instability thresholds. We then analyse the magnetic field power spectra in a frequency range 0.003–10 Hz during four 18-min intervals for different values of β. If β<1, transverse (i.e. Alfvénic) fluctuations are dominant at every frequency. For β≥1, a mixture of compressive (i.e. mirror) and transverse waves is usually observed. For a case with β≃10, there is no frequency where compressive waves are dominant. The values of β and of the proton temperature anisotropy are thus important but not the only parameters which determine the dominant mode, compressive or transverse, at the proton scales in the magnetosheath.