In this paper, we consider nonlinear multidimensional Cahn-Hilliard and Kuramoto-Sivashinsky equations that have many important applications in physics and chemistry, and a certain natural generalization of these two equations to which we refer to as the generalized Cahn-Hilliard-Kuramoto-Sivashinsky equation. For an arbitrary number of spatial independent variables, we present a complete list of cases when the latter equation admits nontrivial local conservation laws of any order, and for each of those cases, we give an explicit form of all the local conservation laws of all orders modulo trivial ones admitted by the equation under study. In particular, we show that the original Kuramoto-Sivashinsky equation admits no nontrivial local conservation laws, and find all nontrivial local conservation laws for the Cahn-Hilliard equation.