© 2015 The Authors. We measure the stellar mass-star formation rate (SFR) relation in star-forming disc galaxies at z ≤ 0.085, using Galaxy Zoo morphologies to examine different populations of spirals as classified by their kiloparsec-scale structure. We examine the number of spiral arms, their relative pitch angle, and the presence of a galactic bar in the disc, and show that both the slope and dispersion of the M*-SFR relation is constant when varying all the above parameters. We also show that mergers (both major and minor), which represent the strongest conditions for increases in star formation at a constant mass, only boost the SFR above the main relation by ~0.3 dex; this is significantly smaller than the increase seen in merging systems at z > 1. Of the galaxies lying significantly above the M*-SFR relation in the local Universe, more than 50 per cent are mergers.We interpret this as evidence that the spiral arms, which are imperfect reflections of the galaxy's current gravitational potential, are either fully independent of the various quenching mechanisms or are completely overwhelmed by the combination of outflows and feedback. The arrangement of the star formation can be changed, but the system as a whole regulates itself even in the presence of strong dynamical forcing.