Fundamental solutions (Green’s functions) to Grad’s steady-state linearised 13-moment equations for non-equilibrium gas flows are derived. The creeping microscale gas flows, to which they pertain, are important to understanding the behaviour of atmospheric particulate and the performance of many potential micro/nano technologies. Fundamental solutions are also derived for the regularised form of the steady-state linearised 13-moment equations, due to Struchtrup & Torrilhon (Phys. Fluids, vol. 15 (9), 2003, pp. 2668–2680). The solutions are compared to their classical and ubiquitous counterpart: the Stokeslet. For an illustration of their utility, the fundamental solutions to Grad’s equations are implemented in a linear superposition approach to modelling external flows. Such schemes are mesh free, and benefit from not having to truncate and discretise an infinite three-dimensional domain. The high accuracy of the technique is demonstrated for creeping non-equilibrium gas flow around a sphere, for which an analytical solution exists for comparison. Finally, to demonstrate the method’s geometrical flexibility, the flow generated between adjacent spheres held at a fixed uniform temperature difference is explored.