  We introduce a new type of local and microlocal asymptotic analysis in
algebras of generalized functions, based on the presheaf properties of those
algebras and on the properties of their elements with respect to a regularizing
parameter. Contrary to the more classical frequential analysis based on the
Fourier transform, we can describe a singular asymptotic spectrum which has
good properties with respect to nonlinear operations. In this spirit we give
several examples of propagation of singularities through nonlinear operators.
