The technique of diffusion tensor tractography utilizes directions of maximum diffusion to reconstruct pathways of white matter structures in the brain. Critically, successful tracking of these white matter pathways depends on well-defined maximal diffusion directional information. By examination of diffusion tensor field properties in the human brain, we demonstrate that the geometry of tracked pathways is influenced by points in the field where the maximum diffusion direction is poorly defined. In common with tensor fields describing other mathematical and physical phenomena, such as 3D surface differential geometry and gravitational fields, we refer to these points as singularities. Here we describe an automated procedure for detecting singularities and demonstrate that these occur where there is (i) fiber crossing, (ii) pathways passing close to one another within a voxel (partial volume effect), and (iii) noise propagation into low anisotropy regions. In order to highlight the relevance of singularities in tracking white matter structures, we determined their effect on computation of the cortico-spinal pathway.