We study the stability of various difference approximations of the Euler--Korteweg equations. This system of evolutionary PDEs is a classical isentropic Euler system perturbed by a dispersive (third order) term. The Euler equations are discretized with a classical scheme (e.g., Roe, Rusanov, or Lax--Friedrichs scheme), whereas the dispersive term is discretized with centered finite differences. We first prove that a certain amount of numerical viscosity is needed for a difference scheme to be stable in the Von Neumann sense. Then we consider the entropy stability of difference approximations. For that purpose, we introduce an additional unknown, the gradient of a function of the density. The Euler--Korteweg system is transformed into a hyperbolic system perturbed by a second order skew symmetric term. We prove entropy stability of Lax--Friedrichs type schemes under a suitable Courant--Friedrichs--Levy condition. In addition, we propose a spatial discretization of the Euler--Korteweg system seen as a Hamil...