We employ neural networks for classification of data of the TUS fluorescence telescope, the world's first orbital detector of ultra-high energy cosmic rays. We focus on two particular types of signals in the TUS data: track-like flashes produced by cosmic ray hits of the photodetector and flashes that originated from distant lightnings. We demonstrate that even simple neural networks combined with certain conventional methods of data analysis can be highly effective in tasks of classification of data of fluorescence telescopes.