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We consider the role of small scale geometric and material features at geometric discontinuities other than a crack and study their relevance to fracture analysis. We are motivated by relatively recent experiments that show that under certain circumstances fracture initiation from geometric (sharp notches) and material (bimaterial interface corners) discontinuities can be successfully correlated with critical values of stress intensities that arise from a linear elastic analysis of the corresponding singular stress state. Implicit to such an approach is the idea that perturbations of the elastic fields near the discontinuity, which of course destroy the singular stresses, occur over a scale that is sufficiently small so that the complex behavior in this region is correlated by the elastic stress intensity. While the fracture mechanician will recognize these ideas as extensions of classical linear elastic fracture mechanics, significant differences exist and these are discussed in detail. We motivate the ideas through the use of a series of model problems in antiplane shear that are mostly amenable to exact analyses. We expect that the ideas carry through, albeit at the expense of far more complicated analysis, to planar and even three-dimensional situations.