  We introduce the notion of reticular Legendrian unfoldings in order to
investigate stabilities of bifurcations of wavefronts generated by a
hypersurface germ with a boundary, a corner, or an r-corner in a smooth n
dimensional manifold. We define several stabilities of reticular Legendrian
unfoldings and prove that they and the stabilities of corresponding generating
families are all equivalent and give the classification of generic bifurcations
of wavefronts in the cases r=0,n<=5 and r=1,n<= 3respectively.
