It is shown that the random electric fields of charge dislocations in insulators should cause appreciable broadening of sharp resonance lines such as spin-resonance signals. The degree of broadening is very sensitive to the screening of the dislocations by charged point defects. Since the shape of the inhomogeneously broadened resonance line monitors the distribution of the internal fields, the lineshape can be used to deduce properties of the charged dislocations, like their charge per unit length and the degree of screening. Quantitative calculations are presented which suggest that spectroscopic methods, notably spin resonance, should provide a convenient method of investigating charged dislocations.