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Conductivity and elastic moduli are related to the second moment of the appropriate field distribution in a disordered material. We argue that breakdown is related to the very high moments of this local-field distribution, and the moment spectrum thus quantifies the crossover between these two very different classes of properties. We first analyze the moment spectrum, and hence the moduli to breakdown crossover, for an isolated cracklike Aaw in electrical, dielectric, and mechanical problems. A result central to this analysis is the form of the electric field or stress distribution due to such a crack. The Appendix collects these scaling forms for a cracklike elliptical (or ellipsoidal) void in electrical and mechanical problems and for a long, thin metallic inclusion in a dielectric and is a major component of the paper. When a distribution of Aaws is present in a material, the moment value at which crossover to extreme scaling occurs is determined by the asymptotic form of the tail of the local load distribution. We discuss the location of this crossover for exponential, algebraic, and multifractal tails. Finally, we discuss the ability of the moment spectrum of the local load distribution to provide nondestructive tests of the presence of large and/or sharp cracklike Aaws in a material. We show that for moment values m )4 it is possible to detect the presence of a sharp crack in electrical and mechanical problems. In three-dimensional dielectrics containing a sharp metal inclusion, it is shown that such detection is possible for m & 3.