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 similar to 2. We start from a stellar-mass-selected sample of galaxies with reliable SI.--42 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 huninosities 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. alpha, where S-v proportional to v(-alpha)) are estimated using their stacked.4(11 GElz 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 similar to 2.0 deg(2). Using this methodology, we constrain the radio spectral index and FRC index of star-forming galaxies with M, > 1010 M and 0 < z < 2.3. We find that alpha(1.4GHz)(610MHz) 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. Delta log(SSFR)(MS) = log[SSFR(galaxy)/SSFRMS(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, qp[R, displays a moderate but statistically significant redshift evolution as q(FIR) (z) = (2.35 +/- 0.08)x(1+z)(-0.12 +/- 0.04), consistent with some previous literature. Finally, we find no significant correlation between CM and Delta log(SSER)(MS), though a weak positive trend, as observed in one of our redshift bins (i.e. Delta[q(FIR)]/Delta[Delta logSSFR)(MS)] = 0.22 +/- 0.07 at 0.5 < z < 0.8), cannot be firmly ruled out using our dataset.