  It has been recently established that heterogeneous bootstrap percolation and
related dynamic facilitation models exhibit a complex hierarchy of continuous
and discontinuous transitions depending on lattice connectivity and kinetic
constraints. Here the range of the previously observed phase diagram topologies
and higher-order singularities is extended to disconnected glass-glass
transitions and to cusp and swallowtail bifurcations (which can be generic and
degenerate). The phase diagram and the order parameter for two different types
of spin mixtures are analytically determined and an experimental realization of
the new predictions emerging in our approach is suggested.
