The motion of oscillatorylike nonlinear Hamiltonian systems, driven by a weak noise, is considered. A general method to find regions of stability in the phase space of a randomly driven system, based on a specific Poincaré map, is proposed and justified. Physical manifestations of these regions of stability are coherent clusters. We illustrate the method and demonstrate the appearance of coherent clusters with two models motivated by the problems of waveguide sound propagation and Lagrangian mixing of passive scalars in the ocean. We find bunches of sound rays propagating coherently in an underwater waveguide through a randomly fluctuating ocean at long distances. We find clusters of passive particles to be advected coherently for a comparatively long time by a random two-dimensional flow modeling mixing around a fixed vortex.