Context. The hydrogen Lyman-alpha (Ly alpha) line, the brightest rest-frame ultraviolet line of high-redshift galaxies, exhibits a large variety of shapes, which is due to factors at different scales, from the interstellar medium to the intergalactic medium (IGM). Aims. The aim of this work is to provide a systematic inventory and classification of the spectral shapes of Ly alpha emission lines to better understand the general population of high-redshift Ly alpha emitting galaxies (LAEs). Methods. Using the unprecedentedly deep data from the MUSE eXtremely Deep Field (MXDF; up to 140 hour exposure time), we selected 477 galaxies observed in the similar to 2.8-6.6 redshift range, 15 of which have a systemic redshift from nebular lines. We developed a method to classify Ly alpha emission lines in four spectral and three spatial categories by combining a pure spectral analysis with a narrow-band image analysis. We measured spectral properties, such as the peak separation and the blue-to-total flux ratio for the double-peaked galaxies. Results. To ensure a robust sample for statistical analysis, we define two unbiased subsets, inclusive and restrictive, by applying thresholds for signal-to-noise ratio, peak separation, and Ly alpha luminosity, yielding a final unbiased sample of 206 galaxies. Our analysis reveals that between 32% and 51% of the galaxies exhibit double-peaked profiles, with peak separations ranging from 150 km s-1 to nearly 1600 km s-1. The fraction of double-peaked galaxies seems to evolve dependently with the Ly alpha luminosity, while we do not see a severe decrease in this fraction with redshift, which is expected given the IGM attenuation at high redshift. An artificial increase in the number of double-peaked galaxies at the highest redshifts may cause the observation of a plateau instead of a decrease. A notable number of these double-peaked profiles show blue-dominated spectra, suggesting unique gas dynamics and inflow characteristics in some high-redshift galaxies. The consequent fraction of blue-dominated spectra needs to be confirmed by obtaining new systemic redshift measurements. Among the double-peaked galaxies, 4% are spurious detections, that is, the blue and red peaks do not come from the same spatial location. Around 20% out of the 477 sources of the parent sample lie in a complex environment, meaning there are other clumps or galaxies at the same redshift within a distance of 30 kpc. Conclusions. Our results suggest that the double-peaked LAE fraction may trace the evolution of IGM attenuation, but the faintest galaxies must be observed at high redshift. We also need more data to confirm the trend seen at low redshift. In addition, it is crucial to obtain secure systemic redshifts for LAEs to better constrain the nature of the Ly alpha double-peaked lines. Statistical samples of double-peaked and triple-peaked galaxies are a promising probe of the evolution of the physical properties of galaxies across cosmic time.