In the Stokesian limit, the streamline topology around a single neutrally buoyant sphere is identical to the topology of pair-sphere pathlines, both in an ambient simple shear flow. In both cases there are fore–aft symmetric open and closed trajectories spatially demarcated by an axisymmetric separatrix surface. We show that the topology of the fluid pathlines around a neutrally buoyant freely rotating spheroid, in simple shear flow, is profoundly different, and will have a crucial bearing on transport from such particles in shearing flows. An inertialess non-Brownian spheroid in a simple shear flow rotates indefinitely in any one of a one-parameter family of Jeffery orbits. The parameter is the orbit constant $C$, with $C = 0$ and $C = \infty$ denoting the limiting cases of a spinning (log-rolling) spheroid, and a spheroid tumbling in the flow–gradient plane, respectively. The streamline pattern around a spinning spheroid is qualitatively identical to that around a sphere regardless of its aspect ratio. For a spheroid in any orbit other than the spinning one ($C >0$), the velocity field being time dependent in all such cases, the fluid pathlines may be divided into two categories. Pathlines in the first category extend from upstream to downstream infinity without ever crossing the flow axis; unlike the spinning case, these pathlines are fore–aft asymmetric, suffering a net displacement in both the gradient and vorticity directions. The second category includes primarily those pathlines that loop around the spheroid, and to a lesser extent those that cross the flow axis, without looping around the spheroid, reversing direction in the process. The residence time, in the neighbourhood of the spheroid, is a smooth function of upstream conditions for pathlines belonging to the first category. In contrast, the number of loops, and thence, the residence time associated with pathlines in the second category, is extremely sensitive to upstream conditions. Plots reveal a fractal structure with singularities distributed on a Cantor-like set, suggesting the existence of a chaotic saddle in the vicinity of the spheroid.