The hypothesis of the self-induced collapse of the inflaton wave function was introduced as a candidate for the physical process responsible for the emergence of inhomogeneity and anisotropy at all scales. In particular, we consider different proposal for the precise form of the dynamics of the inflaton wave function: (i) the GRW-type collapse schemes proposals based on spontaneous individual collapses which generate nonvanishing expectation values of various physical quantities taken as ansatz modifications of the standard inflationary scenario; (ii) the proposal based on a Continuous Spontaneous Localization (CSL) type modification of the Schrödinger evolution of the inflaton wave function, based on a natural choice of collapse operator. ...