We present a systematic numerical investigation of bifurcations in the two-dimensional incompressible Navier-Stokes flow past a confined circular cylinder. The results indicate that there is a qualitative correspondence between changes in the traction profiles of the steady Navier-Stokes equations and bifurcations of the long-time behavior of the unsteady Navier-Stokes equations. The bifurcations include the appearance of symmetry breaking, oscillations, and multiple steady solutions. The well-known planar Schäfer-Turek benchmark is considered with Reynolds numbers up to 1000. For the analysis of bifurcations and traction profiles, several numerical strategies are applied, including a duality method for computing traction profiles, deflation methods, and linear stability analysis. Long-time flow behavior is often explored through direct numerical simulation of the unsteady equations; an approach that is computationally demanding. The relations presented here indicate the possibility of a computationally inexpensive strategy to detect critical Reynolds numbers. This work is in collaboration with J. Blechta, C. Cach and K. Tuma.