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The stellar velocity distribution in the Solar Neighbourhood displays kinematic substructures, which are possibly signatures of the bar and spiral arms of the Milky Way and of previous accretion events. These kinematic substructures -- moving groups -- can be thought of as continuous manifolds in the 6D phase space, and the ridges in the $V_ϕ-R$ and arches in the $V_ϕ-V_R$ plane, discovered with the Gaia mission, as projections of these manifolds. We develop and apply a methodology to perform a blind search for substructure in the Gaia EDR3 6D data, and obtain a sampling of the manifolds. The method consists in the execution of the Wavelet Transform in small volumes of the Milky Way disc, and the grouping of these local solutions into global ones with a method based on the Breadth-first search algorithm from Graph Theory. We reveal the complex skeleton of the velocity distribution, sampling nine main moving groups in a large region of the disc ($6$ kpc, $60$ deg, and $2$ kpc in the radial, azimuthal, and vertical directions). In the radial direction, the groups deviate from lines of constant angular momentum that one would naively expect from first order effect of resonances. The azimuthal velocity of Acturus, Bobylev, and Hercules is non-axisymmetric. For Hercules, we measure an azimuthal gradient of $-0.50$ km/s/deg at $R=8$ kpc. We detect a vertical asymmetry in the azimuthal velocity for Coma Berenices, which is not expected in a resonance of the bar, supporting previous hypothesis of incomplete vertical phase-mixing. When we apply the same methodology to simulations of barred galaxies, we extract substructures corresponding to the Outer Linbdlad and the 1:1 Resonances and observe patterns consistent with the data. This data-driven characterization allows for a quantitative comparison with models, providing a key tool to comprehend the dynamics of the Milky Way. (Abridged)