The most widely used spike-sorting algorithms are semiautomatic in practice, requiring manual tuning of the automatic solution to achieve good performance. In this work, we propose a new fully automatic spike-sorting algorithm that can capture multiple clusters of different sizes and densities. In addition, we introduce an improved feature selection method, by using a variable number of wavelet coefficients, based on the degree of non-Gaussianity of their distributions. We evaluated the performance of the proposed algorithm with real and simulated data. With real data from single-channel recordings, in ~95% of the cases the new algorithm replicated, in an unsupervised way, the solutions obtained by expert sorters, who manually optimized the solution of a previous semiautomatic algorithm. This was done while maintaining a low number of false positives. With simulated data from single-channel and tetrode recordings, the new algorithm was able to correctly detect many more neurons compared with previous implementations and also compared with recently introduced algorithms, while significantly reducing the number of false positives. In addition, the proposed algorithm showed good performance when tested with real tetrode recordings. NEW & NOTEWORTHY We propose a new fully automatic spike-sorting algorithm, including several steps that allow the selection of multiple clusters of different sizes and densities. Moreover, it defines the dimensionality of the feature space in an unsupervised way. We evaluated the performance of the algorithm with real and simulated data, from both single-channel and tetrode recordings. The proposed algorithm was able to outperform manual sorting from experts and other recent unsupervised algorithms.