In the paper, single-fibre models for the extracellular action potential are developed that will allow the potential to the evaluated at an arbitrary field point in the extracellular space. Fourier-domain models are restricted in that they evaluate potentials at equidistant points along a line parallel to the fibre axis. Consequently, they cannot easily evaluate the potential at the boundary nodes of a boundary-element electrode model. The Fourier-domain models employ axial-symmetric ring source models, and thereby provide higher accuracy that the line source model, where the source is lumped into a line source at the centre of the fibre. In the paper new spatial models are developed based on elliptic integrals. These models employ axial-symmetric ring source models, and therefore are more accurate that the line source model. In the analysis, dual transform pairs are identified. Numerical examples including anisotropy show that the spatial models require extreme care in the integration procedure owing to the singularity in the weighting functions. With adequate sampling, the spatial models can evaluate extracellular potentials with high accuracy.