  The nonlinear Schr\"odinger equation (NLSE) stands out as the dispersive
nonlinear partial differential equation that plays a prominent role in the
modeling and understanding of the wave phenomena relevant to many fields of
nonlinear physics. The question of random input problems in the one-dimensional
and integrable NLSE enters within the framework of integrable turbulence, and
the specific question of the formation of rogue waves (RWs) has been recently
extensively studied in this context. The determination of exact analytic
solutions of the focusing 1D-NLSE prototyping RW events of statistical
relevance is now considered as the problem of central importance. Here we
address this question from the perspective of the inverse scattering transform
(IST) method that relies on the integrable nature of the wave equation. We
develop a conceptually new approach to the RW classification in which
appropriate, locally coherent structures are specifically isolated from a
globally incoherent wave train to be subsequently analyzed by implementing a
numerical IST procedure relying on a spatial periodization of the object under
consideration. Using this approach we extend the existing classifications of
the prototypes of RWs from standard breathers and their collisions to more
general nonlinear modes characterized by their nonlinear spectra.
