The Earth's magnetosheath is the region of solar wind which has been decelerated and thermalized at the terrestrial bow shock. The magnetosheath is characterized by high level of turbulent fluctuations covering all the scales from the largest down to the kinetic scales. The turbulent fluctuations are thought to play a fundamental role into key processes such as energy transport and dissipation in plasma. In addition to turbulence, different kind of plasma instabilities are generated in the magnetosheath due to the large temperature anisotropies in the plasma, introduced by its boundaries (bow shock and magnetopause). In this study we use high-quality magnetic field measurements from Cluster spacecraft to investigate the effects of such instabilities on the small-scale turbulence. We show that the steepening of the power spectrum of magnetic field fluctuations in the magnetosheath occurs at the largest characteristic ion scale. However, the spectrum can be modified by the presence of waves/structures at ion scales, shifting the onset of the small-scale (from ion down to electron scales) turbulent cascade towards the smallest ion scale. The small-scale cascade is therefore highly dependent on the presence of kinetic instabilities, waves and local plasma parameters. Here we show that in the absence of strong waves the small-scale turbulence is quasi-isotropic and has a spectral index $\alpha \approx -2.8$. When transverse or compressive waves are present, we observe an anisotropy in the magnetic field components and a decrease in the absolute value of $\alpha$. Slab/2D turbulence also develops in the presence of transverse/compressive waves, resulting in gyrotropy/non-gyrotropy of small-scale fluctuations. The presence of both waves reduces the anisotropy in the amplitude of fluctuations in the small-scale range. In addition, we make use of MMS (Magnetospheric MultiScale) mission data in burst-mode (8192 samples/s) to investigate the nature of electron-scale magnetic turbulence in Earth's magnetosheath and the effects of electron instabilities on its properties.