The minerals processing enterprises are widely using vibrating machines to separate different fractions of materials. Sieving efficiency is greatly dependent on particle trajectories, or orbit, of periodical motion over the sieving decks. A screening process is very dependable on design parameters such as the vibrator power, synchronisation of their drives, and oscillation frequency as well as the stiffness of supporting springs. Deterioration of supporting springs (stiffness reduction and cracks) due to cyclic loading and fatigue is difficult to determine by the visual inspection, static loading tests, or nondestructive testing techniques. Vibration monitoring systems of different vendors are analysed where vibration sensors usually installed on the bearings of vibrators are as well used for supporting springs diagnostics. However, strong cyclic components from the unbalanced exciters and stochastic disturbances from the input stream and vibrating pieces of the material make analysis a not trivial task. The considered vibrating screen is investigated on the 6-DOF (degree-of-freedom) dynamical model to reflect all linear and rotational components of spatial motion. Besides the main periodic motion, the model accounts for stochastic alpha-stable distributed impacts from the material. Instead, the Gaussian normal distribution is considered for the position of equivalent force application point. Supporting springs are represented by the bilinear stiffness characteristics. Specific features of vibration signals (angle of orbit inclination, natural frequency change, harmonics of natural frequency, and phase space plots) are analysed to recognise the weak nonlinear features of a system under conditions of small stiffness changes in springs. The extensive measurements are conducted on the industrial vibrating screen, and the dynamic model is verified by the measurement data. Recommendations are given on failure diagnostics of springs in the industrial vibrating screens.