During individual training on a balance board (BB), misalignment between the ankle joint and the BB axis of rotation may exist. The ankle-pivot misalignment influences the dynamics of human balance and more importantly, the stability properties of the equilibrium positions of the human-BB dynamical system. Similarly, ankle displacement in the upward direction with respect to the BB pivot also plays a critical role in the stability properties of the human-BB system. This paper investigates these effects through bifurcation analyses performed to the ankle-hip model of balance on a BB developed in our previous work. By using local bifurcation analyses, we have obtained the stability regions of the upright posture (UP) of a human-BB model of balance in different parameter spaces. The stability regions are delimited by Hopf, pitchfork, and saddle-node bifurcation curves in some selected parameter spaces. Results show that ankle-pivot misalignment has an impact on the location of the Hopf and unfolds the pitchfork bifurcation curves (found in the aligned case) into saddle-node bifurcation curves. Moreover, ankle-pivot misalignment breaks the mirror symmetry of upright static equilibrium positions and induces the establishment of equilibrium positions away from the vertical UP. With respect to the ankle vertical displacement, it has a minimal impact on the location of the pitchfork bifurcation curves but has a large impact on the location of the Hopf bifurcation curves, especially when combined with large BB time delay. This suggests that a larger ankle vertical displacement may result in sway oscillations of larger amplitude. The analyses also provide insight into different mechanisms of stability that can be found in the ankle-hip model of balance on a BB, namely, limit cycle oscillations and leaning postures. If an individual suffers from neuropathy, results from this study can be useful for researchers and clinicians in understanding what types of instabilities might be encountered, and during rehabilitation how to position the subjects carefully to avoid inadvertent instabilities.