Context. Core-collapse supernovae (CCSNe) are expected to emit gravitational wave signals that could be detected by current and future generation interferometers within the Milky Way and nearby galaxies. The stochastic nature of the signal arising from CCSNe requires alternative detection methods to matched filtering. Aims. We aim to show the potential of machine learning (ML) for multi-label classification of different CCSNe simulated signals and noise transients using real data. We compared the performance of 1D and 2D convolutional neural networks (CNNs) on single and multiple detector data. For the first time, we tested multi-label classification also with long short-term memory (LSTM) networks. Methods. We applied a search and classification procedure for CCSNe signals, using an event trigger generator, the Wavelet Detection Filter (WDF), coupled with ML. We used time series and time-frequency representations of the data as inputs to the ML models. To compute classification accuracies, we simultaneously injected, at detectable distance of 1 kpc, CCSN waveforms, obtained from recent hydrodynamical simulations of neutrino-driven core-collapse, onto interferometer noise from the O2 LIGO and Virgo science run. Results. We compared the performance of the three models on single detector data. We then merged the output of the models for single detector classification of noise and astrophysical transients, obtaining overall accuracies for LIGO (~99%) and (~80%) for Virgo. We extended our analysis to the multi-detector case using triggers coincident among the three ITFs and achieved an accuracy of ~98%.