Permanent magnet synchronous motors (PMSMs) are becoming more popular, both in industrial applications and in electric and hybrid vehicle drives. Unfortunately, like the others, these are not reliable drives. As in the drive systems with induction motors, the rolling bearings can often fail. This paper focuses on the possibility of detecting this type of mechanical damage by analysing mechanical vibrations supported by shallow neural networks (NNs). For the extraction of diagnostic symptoms, the Fast Fourier Transform (FFT) and the Hilbert transform (HT) were used to obtain the envelope signal, which was subjected to the FFT analysis. Three types of neural networks were tested to automate the detection process: multilayer perceptron (MLP), neural network with radial base function (RBF), and Kohonen map (self-organizing map, SOM). The input signals of these networks were the amplitudes of harmonic components characteristic of damage to bearing elements, obtained as a result of FFT or HT analysis of the vibration acceleration signal. The effectiveness of the analysed NN structures was compared from the point of view of the influence of the network architecture and various parameters of the learning process on the detection effectiveness.