Context. A multitude of spectral activity indicators are routinely computed nowadays from the spectra generated as part of planet-hunting radial velocity surveys. Searching for shared periods among them can help to robustly identify astrophysical quantities of interest, such as the stellar rotation period. However, this identification can be complicated due to the fact that many different peaks occur in the periodograms. This is especially true in the presence of aliasing and spurious signals caused by environmental influences affecting the instrument. Aims. Our goal is to test a clustering algorithm to find signals with the same periodicity, (i.e. with the stellar rotation period) in the periodograms of a large number of activity indicators. On this basis, we have looked to evaluate the correlations between activity indicators and fundamental stellar parameters. Methods. We used generalised Lomb–Scargle periodograms to find periodic signals in 24 activity indicators, spanning the VIS and NIR channels of the CARMENES spectrograph. Common periods were subsequently determined by a machine learning algorithm for density-based spatial clustering of applications with noise (DBSCAN). Results. The clustering analysis of the signals apparent in the spectral activity indicators is a powerful tool for the detection of stellar rotation periods. It is straightforward to implement and can be easily automated, so that large data sets can be analysed. For a sample of 136 stars, we were able to recover the stellar rotation period in a total of 59 cases, including 3 with a previously unknown rotation period. In addition, we analysed spurious signals frequently occurring at the period of one year and its integer fractions, concluding that they are likely aliases of one underlying signal. Furthermore, we reproduced the results of several previous studies on the relationships between activity indicators and the stellar characteristics.