We define bifurcations of control-affine systems in the plane and classify all generic 1-parameter bifurcations at control-regular points. More precisely, we classify topological bifurcations of invariants of usual feedback equivalence. Such bifurcations form six different classes: two bifurcations of equilibrium sets, two bifurcations of critical sets, and two bifurcations of pairs of invariants. We also classify all generic 1-parameter families of control-affine systems with respect to orbital feedback equivalence.