We report on the rheo-optical properties of dilute surfactant solutions which exhibit a shear-induced thickening transition. The surfactant investigated is cetyltrimethylammonium tosylate at concentrations around 0.2 wt % in water. By comparing data obtained by rheology and rheo-optics, we first confirm that the thickening transition is associated with the transformation of the initially isotropic micellar liquid into a birefringent and strongly aligned phase. In both experiments, the transition occurs at the same critical shear rate, denoted gamma (over dot)(c). Using a new treatment for the analysis of the rheo-optical data, we have been able to determine the birefringence Deltan and extinction angle chi at steady state, but also as a function of the time and of the spatial coordinate in the 1 mm gap of a Couette-type shearing cell (with a spatial resolution of 15 mum). Starting from a solution in a quiescent state and applying steady shear, we provide an accurate description of the kinetics of formation of the shear-induced and viscous phase. At steady state, we observe strong variations of the flow birefringence An as a function of the spatial coordinates within the gap, as well as a failure of the stress-optical law. The extinction angle chi is then close to 0degrees, independent of the shear rate and of the spatial coordinates. The flow birefringence also exhibits rapid fluctuations around the average profiles, on a length scale comprised between 50 and 100 mum. These fluctuations are interpreted as shear bands of enhanced and reduced birefringence, a result which suggests a strongly inhomogeneous and instable shear flow in the shear-induced phase.