Using 2D particle-in-cell (PIC) simulations coupled to a fluid description of the gas dynamics, we study the electrostatic instabilities developing in the axial–azimuthal plane of a Hall thruster, during several periods of a low-frequency oscillation (the so-called breathing mode at [Formula: see text]). As done in experiments, the 2D PIC-MCC (Monte Carlo collision) code is coupled to an electrical circuit in order to partially damp the (otherwise large) discharge current fluctuations at the breathing mode frequency. The different electrostatic higher frequency modes that develop in the plasma are analyzed using a two-point power spectral density reconstruction method, which allows us to generate the dispersion diagrams (in the frequency-wavenumber space) along the axial and azimuthal directions and at different times during the low-frequency breathing mode oscillations. This technique allows us to distinguish between different well-identified instabilities: the electron cyclotron drift instability and its evolution toward an ion acoustic wave and the ion transit time instability. These instabilities are usually considered unidirectional (either axial or azimuthal); however, it is shown here that they exist in both directions. This two-dimensional character is instrumental in understanding where these instabilities grow and how they propagate in the thruster channel and plume. A theoretical discussion of this aspect is proposed in Paper II. The effects of (i) the azimuthal length of the simulation box and (ii) the electron temperature injection at the cathode are also discussed.