The first JWST deep surveys have expanded our understanding of the morphological evolution of galaxies across cosmic time. The improved spatial resolution and near-infrared (NIR) coverage have revealed a population of morphologically evolved galaxies at very early epochs. However, all previous works are based on relatively small samples; this has prevented accurate probing of the morphological diversity at cosmic dawn. Leveraging the wide area coverage of the COSMOS-Web survey, we quantified the abundance of different morphological types from z∼7 with unprecedented statistics and established robust constraints on the epoch of emergence of the Hubble sequence. We measured the global morphologies (spheroids, disk-dominated, bulge-dominated, peculiar) and resolved morphologies (stellar bars) for about 400,000 galaxies down to F150W=27 using deep learning; this represents an increase of two orders of magnitude over previous studies. We provide reference stellar mass functions (SMFs) of different morphologies between z∼0.2 and z∼7 as well as best-fit parameters to inform models of galaxy formation. All catalogs and data are made publicly available. At redshift ( z > 4.5 ), the massive galaxy population (łog M_/M_⊙>10) is dominated by disturbed morphologies (( ∼70% )), even in the optical rest frame, and very compact objects (( ∼30% )) with effective radii smaller than ( ∼500 pc ). This confirms that a significant fraction of the star formation at cosmic dawn occurs in very dense regions, although the stellar mass for these systems could be overestimated. Galaxies with Hubble-type morphologies, including bulge- and disk-dominated galaxies, arose rapidly around ( z ∼ 4 ) and dominate the morphological diversity of massive galaxies as early as ( z ∼ 3 ). Using stellar bars as a proxy, we speculate that stellar disks in massive galaxies might have been common ($>50%$) among the star-forming population since cosmic noon (( z ∼ 2 2.5 )) and formed as early as z∼7. Massive quenched galaxies are predominantly bulge-dominated from ( z ∼ 4 ) onward, suggesting that morphological transformations briefly precede or are simultaneous to quenching mechanisms at the high-mass end. Low-mass (łog M_/M_⊙<10) quenched galaxies are typically disk-dominated, which points to different quenching routes at the two ends of the stellar mass spectrum from cosmic dawn.