In this work we study the Boussinesq system, which is a natural model for the propagation of long waves on the surface of water with a small amplitude and is used to compute a complete set of local conservation laws of the model through the direct method. In this method, some local multipliers are found to construct the fluxes of the conservation law. These multipliers are used to find new conservation laws via another method such as Noether's theorem, Boyer's formulation, Homotopy operator method and Ibragimov's theorem. It is noteworthy that this paper reviews these methods to compare all obtained fluxes and local conservation laws.