The method of regularised stokeslets is widely used in microscale biological fluid dynamics due to its ease of implementation, natural treatment of complex moving geometries, and removal of singular functions to integrate. The standard implementation of the method is subject to high computational cost due to the coupling of the linear system size to the numerical resolution required to resolve the rapidly- varying regularised stokeslet kernel. Here we show how Richardson extrapolation with coarse values of the regularisation parameter is ideally-suited to reduce the quadrature error, hence dramatically reducing the storage and solution costs without loss of accuracy. Numerical experiments on the resistance and mobility problems in Stokes flow support the analysis, confirming several orders of magnitude improvement in accuracy and/or efficiency. This Zenodo repository contains all the MATLAB code for the manuscript: Gallagher and Smith 2021, The art of coarse Stokes: Richardson extrapolation improves the accuracy and efficiency of the method of regularized stokeslets. This work was supported by Engineering and Physical Sciences Research Council (EPSRC) Award No. EP/N021096/1. MTG acknowledges support from EPSRC Centre Grant EP/N014391/2.