With an unprecedented astrometric and photometric data precision, Gaia EDR3 provides, for the first time, the opportunity to systematically detect and map, in the optical bands, the low-mass populations of the star forming regions (SFRs) in the Milky Way. We aim to provide a catalogue of the Gaia EDR3 data (photometry, proper motions and parallaxes) of the young stellar objects (YSOs) identified in the Galactic plane |b|<30{deg}) within about 1.5kpc. The catalogue of the SFRs to which they belong is also provided to study the properties of the very young clusters and put them in the context of the Galaxy structure. We applied the machine learning unsupervised clustering algorithm density-based spatial clustering of applications with noise (DBSCAN) to a sample of Gaia EDR3 data photometrically selected on the region where very young stars (t<~10Myr) are expected to be found, with the aim of identifying co-moving and spatially consistent stellar clusters. A sub-sample of 52 clusters, selected among the 7323 found with DBSCAN, has been used as template data set to identify very young clusters from the pattern of the observed colour-absolute magnitude diagrams through a pattern-match process. We find 124440 candidate YSOs clustered in 354 SFRs and stellar clusters younger than 10Myr and within <~1.5kpc. In addition, 65 863 low-mass members of 322 stellar clusters located within ~500pc and with ages 10Myr<~t<~100Myr were also found. The selected YSOs are spatially correlated with the well-known SFRs. Most of them are associated with well-concentrated regions or complex structures of the Galaxy, and a substantial number of them have been recognised for the first time. The massive SFRs, such as, for example, Orion, Sco-Cen, and Vela, located within 600-700 pc trace a very complex three-dimensional pattern, while the farthest ones seem to follow a more regular pattern along the Galactic plane.