  The morphology of galaxies gives essential constraints on the models of
galaxy evolution. The morphology of the features in the low-surface-brightness
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 (ETGs) from the MATLAS survey. The images reach a surface-brightness
limit of $28.5-29$ mag 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 disks, and simultaneously noted the presence of
contaminating sources, such as Galactic cirrus. We also inspected internal
sub-structures 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$_\odot$, 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.
