Population vulnerability to an infectious disease epidemic is commonly summarized by system-level indicators such as the epidemic threshold: a single critical boundary. Yet transmission is heterogeneous across communities, host groups and transmission pathways, and public-health decisions often require identifying which parts of the system become vulnerable, and under which conditions. Here we show that epidemic criticality can itself be fragmented across population structure. Using multitype branching processes, we identify singularities governing the expected size of outbreaks that ultimately become extinct and show that coupling between population strata transforms their individual thresholds into complex-valued critical points. Their real parts locate critical changes along the transmissibility axis, their imaginary parts determine their strength and smearing, and their modes identify the subpopulations involved. In Italy, this framework reveals localized vulnerability to respiratory-pathogen emergence and improves vaccine allocation over importation-based strategies. For measles in Texas, it identifies spatial units more homogeneous in observed outbreak burden than standard administrative or metropolitan partitions. In a One Health model of livestock-associated MRSA, it separates occupational and human--animal transmission pathways. Epidemic vulnerability is therefore organized by a structured critical landscape rather than a single threshold, and resolving this landscape can directly inform public-health risk assessment and intervention.