Over the past decades, the lattice Boltzmann method (LBM) has enjoyed a growing popularity in computational fluid dynamics. Its mesoscospic description based on kinetic theory of gases makes it very different from standard Navier-Stokes based solvers, which results in a simple, very efficient and highly parallelizable algorithm together with low dissipation properties. However, the standard scheme, based on a simple approximation of the Boltzmann collision (BGK model), is restricted to simulations of isothermal and weakly compressible flows for which the Mach number cannot be too large (Ma < 0.4). This is mainly due to time and space discretization errors in the numerical scheme. In order to highlight the origin of the stability issues, it is proposed in this work to perform linear analyses of the LBM in the sense of von Neumann. Thanks to an original interpretation of the eigenvectors of the linear systems, so-called non-hydrodynamic modes can be exhibited as inherent to the mesoscopic fluid description. The role of these modes in the numerical issues (stability, spurious noise) is further investigated, as well as possible ways to efficiently damp them in a simulation, involving more sophisticated collision models.