Conductivity fluctuations and the closely related weakly nonlinear conductivity of a composite conductor are examined for a variety of local microgeometries, some of which have been used in the past in models of continuum percolation. A great sensitivity to microgeometric details has been found that had not been previously appreciated. This is invoked to explain the failure of a conventional approach to implementing a self-consistent effective-medium approximation for conductivity fluctuations. It is suggested that measurements of the noise or of weak nonlinearities in electrical or mechanical behavior can be used to get information about microgeometric details of a composite to which the usual conductivity or elastic stiffness moduli are insensitive. In particular, parameters that affect the breakdown properties seem to be accessible to such nondestructive analysis.