We investigate, theoretically and by analog experiment, the distribution of paths for large fluctuations away from a stable state. We have found critical broadening of the distribution of the paths coming to a cusp point that represents the simplest generic singularity in the pattern of most probable (optimal) fluctuational paths in nonequilibrium systems. The critical behavior can be described by a Landau-type theory. We predict and observe two-ridged distributions for arrivals on a switching line that separates the areas reached along optimal paths of different types.