A novel framework for joint detection of sleep spindles and K-complex events, two hallmarks of sleep stage S2, is proposed. Sleep electroencephalography (EEG) signals are split into oscillatory (spindles) and transient (K-complex) components. This decomposition is conveniently achieved by applying morphological component analysis (MCA) to a sparse representation of EEG segments obtained by the recently introduced discrete tunable Q-factor wavelet transform (TQWT). Tuning the Q-factor provides a convenient and elegant tool to naturally decompose the signal into an oscillatory and a transient component. The actual detection step relies on thresholding (i) the transient component to reveal K-complexes and (ii) the time-frequency representation of the oscillatory component to identify sleep spindles. Optimal thresholds are derived from ROC-like curves (sensitivity vs. FDR) on training sets and the performance of the method is assessed on test data sets. We assessed the performance of our method using full-night sleep EEG data we collected from 14 participants. In comparison to visual scoring (Expert 1), the proposed method detected spindles with a sensitivity of 83.18% and false discovery rate (FDR) of 39%, while K-complexes were detected with a sensitivity of 81.57% and an FDR of 29.54%. Similar performances were obtained when using a second expert as benchmark. In addition, when the TQWT and MCA steps were excluded from the pipeline the detection sensitivities dropped down to 70% for spindles and to 76.97% for K-complexes, while the FDR rose up to 43.62 and 49.09%, respectively. Finally, we also evaluated the performance of the proposed method on a set of publicly available sleep EEG recordings. Overall, the results we obtained suggest that the TQWT-MCA method may be a valuable alternative to existing spindle and K-complex detection methods. Paths for improvements and further validations with large-scale standard open-access benchmarking data sets are discussed.