The morphology of galaxies gives essential constraints on the models of galaxy evolution. The morphology of the features in the low-surface-brightness (LSB) regions of galaxies has not been fully explored yet because of observational difficulties. Here we present the results of our visual inspections of very deep images of a large volume-limited sample of 177 nearby massive early-type galaxies from the MATLAS survey. The images reach a surface-brightness limit of 28.5-29mag/arcsec^2^ in the g' band. Using a dedicated navigation tool and questionnaire, we looked for structures at the outskirts of the galaxies such as tidal shells, streams, tails, disturbed outer isophotes, or peripheral star-forming discs, and simultaneously noted the presence of contaminating sources, such as Galactic cirrus. We also inspected internal substructures such as bars and dust lanes. We discuss the reliability of this visual classification investigating the variety of answers made by the participants. We present the incidence of these structures and the trends of the incidence with the mass of the host galaxy and the density of its environment. We find an incidence of shells, stream, and tails of approximately 15%, about the same for each category. For galaxies with masses over 10^11^M_{sun}_, the incidence of shells and streams increases about 1.7 times. We also note a strong unexpected anticorrelation of the incidence of Galactic cirrus with the environment density of the target galaxy. Correlations with other properties of the galaxies, and comparisons to model predictions, will be presented in future papers.