Electrostatic two-stream instabilities play essential roles in an electrostatic collisionless shock formation. They are a key dissipation mechanism and result in ion heating and acceleration. Since the number and energy of the shock-accelerated ions depend on the instabilities, precise identification of the active instabilities is important. Two-dimensional particle-in-cell simulations in a multicomponent plasma reveal ion reflection and acceleration at the shock front, excitation of a longitudinally propagating electrostatic instability due to a nonoscillating component of the electrostatic field in the upstream region of the shock, and generation of up- and down-shifted velocity components within the expanding-ion components. A linear analysis of the instabilities for a C_{2}H_{3}Cl plasma using the one-dimensional electrostatic plasma dispersion function, which includes electron and ion temperature effects, shows that the most unstable mode is the electrostatic ion-beam two-stream instability (IBTI), which is weakly dependent on the existence of electrons. The IBTI is excited by velocity differences between the expanding protons and carbon-ion populations. There is an electrostatic electron-ion two-stream instability with a much smaller growth rate associated with a population of protons reflecting at the shock. The excitation of the fast-growing IBTI associated with laser-driven collisionless shock increases the brightness of a quasimonoenergetic ion beam.