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Following the success of the Plateau de Bure high-z Blue Sequence Survey (PHIBSS), we present the PHIBSS2 legacy program, a survey of the molecular gas properties of star-forming galaxies on and around the star formation main sequence (MS) at different redshifts using NOEMA. This survey significantly extends the existing sample of star-forming galaxies with CO molecular gas measurements, probing the peak epoch of star formation (z=1-1.6) as well as its building-up (z=2-3) and winding-down (z=0.5-0.8) phases. The targets are drawn from the GOODS, COSMOS, and AEGIS deep fields and uniformly sample the MS in the stellar mass (M*) - star formation rate (SFR) plane with log(M*/Msun) = 10-11.8. We describe the survey strategy and sample selection before focusing on the results obtained at z=0.5-0.8, where we report 60 CO(2-1) detections out of 61 targets. We determine their molecular gas masses and separately obtain disc sizes and bulge-to-total (B/T) luminosity ratios from HST I-band images. The median molecular gas-to-stellar mass ratio, gas fraction, and depletion time as well as their dependence with M* and offset from the MS follow published scaling relations for a much larger sample of galaxies spanning a wider range of redshifts. The galaxy-averaged Kennicutt-Schmidt (KS) relation between molecular gas and SFR surface densities is strikingly linear, pointing towards similar star formation timescales within galaxies at any given epoch. In terms of morphology, the molecular gas content, the SFR, the disc stellar mass, and the disc molecular gas fraction do not seem to correlate with B/T and the stellar surface density, which suggests an ongoing supply of fresh molecular gas to compensate for the build-up of the bulge. Our measurements do not yield any significant variation of the depletion time with B/T and hence no strong evidence for morphological quenching within the scatter of the MS., Comment: Accepted for publication in A&A (28 pages, 9 figures, 4 tables)