We study the evolution of the radio spectral index and far-infrared/radio correlation (FRC) across the star-formation rate -stellar masse (i.e. SFR-M * ) plane up to z ∼ 2. We start from a stellar-mass-selected sample of galaxies with reliable SFR and redshift estimates. We then grid the SFR-M * plane in several redshift ranges and measure the infrared luminosity, radio luminosity, radio spectral index, and ultimately the FRC index (i.e. q FIR ) of each SFR-M * -z bin. The infrared luminosities of our SFR-M * -z bins are estimated using their stacked far-infrared flux densities inferred from observations obtained with the Herschel Space Observatory. Their radio luminosities and radio spectral indices (i.e. α, where S ν ∝ ν −α ) are estimated using their stacked 1.4 GHz and 610 MHz flux densities from the Very Large Array and Giant Metre-wave Radio Telescope, respectively. Our far-infrared and radio observations include the most widely studied blank extragalactic fields -GOODS-N, GOODS-S, ECDFS, and COSMOS -covering a total sky area of ∼2.0 deg 2 . Using this methodology, we constrain the radio spectral index and FRC index of star-forming galaxies with M * > 10 10 M and 0 < z < 2.3. We find that α 1.4 GHz 610 MHz does not evolve significantly with redshift or with the distance of a galaxy with respect to the main sequence (MS) of the SFR-M * plane (i.e. Δlog(SSFR) MS = log[SSFR(galaxy)/SSFR MS (M * , z)]). Instead, star-forming galaxies have a radio spectral index consistent with a canonical value of 0.8, which suggests that their radio spectra are dominated by non-thermal optically thin synchrotron emission. We find that the FRC index, q FIR , displays a moderate but statistically significant redshift evolution as q FIR (z) = (2.35±0.08)×(1+z) −0.12 ± 0.04 , consistent with some previous literature. Finally, we find no significant correlation between q FIR and Δlog(SSFR) MS , though a weak positive trend, as observed in one of our redshift bins (i.e. Δ[q FIR ]/Δ[Δlog(SSFR) MS ] = 0.22 ± 0.07 at 0.5 < z < 0.8), cannot be firmly ruled out using our dataset.