We investigate when the freeness of a divisor (V, 0) is inherited by the discriminants for the versal deformations of nonlinear sections of V. We introduce Morse-type singularities for sections and give a criterion for freeness of the discriminant in terms of (V, 0) generically having Morse-type singularities. This criterion is applied to determine when the bifurcation sets of mappings and smoothings of space curves and complete intersections are free. It also explains the failure of freeness for discriminantal arrangements of hyperplanes.