The paper considers scenarios of appearance of rare dangerous attractors [1-4] in typical dissipative oscillation systems with one degree of freedom, nonlinear elastic characteristic of hard type and harmonic excitation. Two types of elastic symmetric nonlinear suspension are examined: with trilinear characteristic and with cubic characteristic of Duffing’s type. The bifurcations diagrams with rare attractors and their domains of attraction are plotted using NLO [5] and SPRING [6] software. For trilinear systems the exact analytical solutions of rare attractors and the exact monodromy matrixes for stability evaluation have found as well [7]. Basins of attraction have built and their comparison also has done for two systems with different parameters. It is shown that rare attractors may be dangerous. We plane to do natural experimental investigation for mechanical driven systems with rare attractors.