Context. The distribution of member stars in the surroundings of an open cluster (OC) can shed light on the process of its formation, evolution, and dissolution. The analysis of structural parameters of OCs as a function of their age and position in the Galaxy constrains theoretical models of cluster evolution. The Gaia catalog is very appropriate for finding members of OCs at large distance from their centers.Aims. We revisit the membership lists of OCs from the solar vicinity, in particular, by extending these membership lists to the peripheral areas through Gaia EDR3. We then take advantage of these new member lists to study the morphological properties and the mass segregation levels of the clusters.Methods. We used the clustering algorithm HDBSCAN on Gaia parallaxes and proper motions to systematically search for members up to 50 pc from the cluster centers. We fit a King’s function on the radial density profile of these clusters and a Gaussian mixture model (GMM) on their two-dimensional member distribution to study their shape. We also evaluated the degree of mass segregation of the clusters and the correlations of these parameters with the age and Galactic position of the clusters.Results. Our method performs well on 389 clusters out of the 467 clusters we selected, including several recently discovered clusters that were poorly studied until now. We report the detection of vast coronae around almost all the clusters and report the detection of 71 OCs with tidal tails. This multiplies the number of these structures that are identified by more than four. The size of the cores is smaller for old clusters than for young ones on average. Moreover, the overall size of the clusters seems to increase slightly with age, but the fraction of stars in the halo seems to decrease. As expected, the mass segregation is more pronounced in the oldest clusters, but no clear trend with age is evident.Conclusions. OCs are more extended than previously expected, regardless of their age. The decrease in the proportion of stars populating the clusters halos highlights the different cluster evaporation processes and the short timescales they need to affect the clusters. Reported parameters such as cluster sizes or mass segregation levels all depend on cluster ages, but cannot be described as single functions of time.Read less <