This paper presents a procedure for the application of a multifractal spectrum to analyse vibration signals obtained during static bending tests of selected fibre reinforced polymer matrix composite materials. The analyses were designed to determine the failure course of the composites tested under bending conditions. Non-stationary changes occurring in the signals include low- and high-energy symptoms of progressive damage in materials, which are difficult to identify and interpret, especially in the early stages. Determined using fractal leaders, the multifractal spectra of the vibration signals calculated for the different stages of the bending test allowed qualitative identification of the changes caused by progressive damage with different energy responses. In addition, the quantitative measures of change in the characteristics of the multifractal spectrum determined clearly indicated the changes occurring in the recorded signals. It can be concluded from the research that the proposed signal processing method based on a multifractal spectrum determined from non-stationary vibration signals is sensitive to changes occurring in these signals.