We present a first statistical study of subproton and electron scales turbulence in the terrestrial magnetosheath using waveform data measured by the Cluster/STAFF Search Coil Magnetometer (SCM) in the frequency range [1, 180] Hz. It is found that clear spectral breaks exist near the electron scale, which separate two power-law like frequency bands referred to as the dispersive and the electron dissipation ranges. The frequencies of the breaks fb are shown to be well correlated with the electron gyroscale rhoe rather than with the electron inertial length de. The distribution of the slopes below fb is found to be narrow and peaks near -2.9, while that of the slopes above fb is found broader, peaks near -5.2 and has values as low as -7.5. This is the first time that such steep power-law spectra are reported in space plasma turbulence. These observations provide strong constraints on theoretical modeling of kinetic turbulence and dissipation in collisionless magnetized plasmas.