  Active polar fluids exhibit spontaneous flow when sufficient active stress is
generated by internal molecular mechanisms. This is also referred to as an
active Fr\'{e}edericksz transition. Experiments have revealed the existence of
competing in-plane and out-of-plane instabilities in three-dimensional active
matter. So far, however, a theoretical model reconciling all observations is
missing. In particular, the role of boundary conditions in these instabilities
still needs to be explained. Here, we characterize the spontaneous flow
transition in a symmetry-preserving three-dimensional active Ericksen-Leslie
model, showing that the boundary conditions select the emergent behavior. Using
nonlinear numerical solutions and linear perturbation analysis, we explain the
mechanism for both in-plane and out-of-plane instabilities under extensile
active stress for perpendicular polarity anchoring at the boundary, whereas
parallel anchoring only permits in-plane flows under contractile stress or
out-of-plane wrinkling under extensile stress.
