Bio-inspired computing using artificial spiking neural networks promises performances outperforming currently available computational approaches. Yet, the number of applications of such networks remains limited due to the absence of generic training procedures for complex pattern recognition, which require the design of dedicated architectures for each situation. We developed a spike-timing-dependent plasticity (STDP) spiking neural network (SSN) to address spike-sorting, a central pattern recognition problem in neuroscience. This network is designed to process an extracellular neural signal in an online and unsupervised fashion. The signal stream is continuously fed to the network and processed through several layers to output spike trains matching the truth after a short learning period requiring only few data. The network features an attention mechanism to handle the scarcity of action potential occurrences in the signal, and a threshold adaptation mechanism to handle patterns with different sizes. This method outperforms two existing spike-sorting algorithms at low signal-to-noise ratio (SNR) and can be adapted to process several channels simultaneously in the case of tetrode recordings. Such attention-based STDP network applied to spike-sorting opens perspectives to embed neuromorphic processing of neural data in future brain implants.