As a part of the CALYPSO large programme, we constrain the properties of protostellar jets and outflows in a sample of 21 Class 0 protostars with internal luminosities, Lint, from 0.035 to 47 Lsun. We analyse high angular resolution (~0.5-1) IRAM PdBI observations in CO (2-1), SO ($5_6-4_5$), and SiO (5-4). CO (2-1), which probes outflowing gas, is detected in all the sources (for the first time in SerpS-MM22 and SerpS-MM18b). Collimated high-velocity jets in SiO (5-4) are detected in 67% of the sources (for the first time in IRAS4B2, IRAS4B1, L1448-NB, SerpS-MM18a), and 77% of these also show jet/outflow emission in SO ($5_6-4_5$). In 5 sources (24% of the sample) SO ($5_6-4_5$) probes the inner envelope and/or the disk. The CALYPSO survey shows that the outflow phenomenon is ubiquitous and that the detection rate of high-velocity jets increases with protostellar accretion, with at least 80% of the sources with Lint>1 Lsun driving a jet. The protostellar flows exhibit an onion-like structure, where the SiO jet (opening angle ~10$^o$) is nested into a wider angle SO (~15$^o$) and CO (~25$^o$) outflow. On scales >300 au the SiO jets are less collimated than atomic jets from Class II sources (~3$^o$). Velocity asymmetry between the two jet lobes are detected in one third of the sources, similarly to Class II atomic jets, suggesting that the same launching mechanism is at work. Most of the jets are SiO rich (SiO/H2 from >2.4e-7 to >5e-6), which indicates efficient release of >1%-10% of silicon in gas phase likely in dust-free winds, launched from inside the dust sublimation radius. The mass-loss rates (from ~7e-8 to ~3e-6 Msun/yr) are larger than what was measured for Class II jets. Similarly to Class II sources, the mass-loss rates are ~1%-50% of the mass accretion rates suggesting that the correlation between ejection and accretion in young stars holds from 1e4 yr up to a few Myr.
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