Context. 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. Aims. 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. Methods. 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. Results. At redshift z > 4.5, the massive galaxy population (log 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 (log 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.