In this paper, we derive scaling relations between photometric observable quantities and disc galaxy rotation velocity Vrot or Tully–Fisher relations (TFRs). Our methodology is dictated by our purpose of obtaining purely photometric, minimal-scatter estimators of Vrot applicable to large galaxy samples from imaging surveys. To achieve this goal, we have constructed a sample of 189 disc galaxies at redshifts z < 0.1 with long-slit Hα spectroscopy from Pizagno et al. and new observations. By construction, this sample is a fair subsample of a large, well-defined parent disc sample of ∼170 000 galaxies selected from the Sloan Digital Sky Survey Data Release 7 (SDSS DR7). The optimal photometric estimator of Vrot we find is stellar mass M★ from Bell et al., based on the linear combination of a luminosity and a colour. Assuming a Kroupa initial mass function (IMF), we find: log [V80/(km s−1)] = (2.142 ± 0.004) + (0.278 ± 0.010)[log (M★/M⊙) − 10.10], where V80 is the rotation velocity measured at the radius R80 containing 80 per cent of the i-band galaxy light. This relation has an intrinsic Gaussian scatter dex and a measured scatter σmeas= 0.056 dex in log V80. For a fixed IMF, we find that the dynamical-to-stellar mass ratios within R80, (Mdyn/M★)(R80), decrease from approximately 10 to 3, as stellar mass increases from M★≈ 109 to 1011 M⊙. At a fixed stellar mass, (Mdyn/M★)(R80) increases with disc size, so that it correlates more tightly with stellar surface density than with stellar mass or disc size alone. We interpret the observed variation in (Mdyn/M★)(R80) with disc size as a reflection of the fact that disc size dictates the radius at which Mdyn/M★ is measured, and consequently, the fraction of the dark matter ‘seen’ by the gas at that radius. For the lowest M★ galaxies, we find a positive correlation between TFR residuals and disc sizes, indicating that the total density profile is dominated by dark matter on these scales. For the highest M★ galaxies, we find instead a weak negative correlation, indicating a larger contribution of stars to the total density profile. This change in the sense of the correlation (from positive to negative) is consistent with the decreasing trend in (Mdyn/M★)(R80) with stellar mass. In future work, we will use these results to study disc galaxy formation and evolution and perform a fair, statistical analysis of the dynamics and masses of a photometrically selected sample of disc galaxies.