Results of numerical investigations, based on full dynamic equations, are presented for wave breaking in a one-dimensional environment with a wave spectrum. The breaking is defined as a process of irreversible collapse of an individual wave in physical space, and the incipient breaker is a wave that reached a dynamic condition of the limiting stability where the collapse has not started yet but is inevitable. The main attention is paid to documenting the evolution of different wave characteristics before the breaking commences. It is shown that the breaking is a localized process that rapidly develops in space and time. No single characteristic, such as wave steepness, wave height, and asymmetry, can serve as a predictor of the incipient breaking. The process of breaking is intermittent; it happens spontaneously and is individually unpredictable. The evolution of geometric, kinematic, and dynamic characteristics of the breaking wave describes the process of breaking itself rather than indicating an imminent breaking. It is shown that the criterion of breaking, valid for the breaking due to modulation instability in one-dimensional waves trains, is not universal if applied to the conditions of spectral environment. In this context, the development of algorithms for parameterization of breaking for wave prediction models and for direct wave simulations is more important.