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The 'positive allometry hypothesis' predicts that ornaments and weapons of sexual selection will scale steeply when among-individual variation in trait size is compared with variation in overall body size. Intuitive and striking, this idea has been explored in hundreds of contemporary animal species and sparked controversy in palaeobiology over the function of exaggerated structures in dinosaurs and other extinct lineages. Recently, however, challenges to this idea have raised questions regarding the validity of the hypothesis. We address this controversy in two ways. First, we suggest the positive allometry hypothesis be applied only to morphological traits that function as visual signals of individual body size. Second, because steep scaling slopes make traits better signals than other body parts, we propose that tests of the positive allometry hypothesis compare the steepness of the scaling relationships of focal, putative signal traits to those of other body parts in the same organism (rather than to an arbitrary slope of 1). We provide data for a suite of 29 extreme structures and show that steep scaling relationships are common when structures function as signals of relative body size, but not for comparably extreme structures that function in other contexts. We discuss these results in the context of animal signalling and sexual selection, and conclude that patterns of static scaling offer powerful insight into the evolution and function of disproportionately large, or extreme, animal structures. Finally, using data from a ceratopsid dinosaur and a pterosaur, we show that our revised test can be applied to fossil assemblages, making this an exciting and powerful method for gleaning insight into the function of structures in extinct taxa. Highlights • Sexually selected signals scale steeply compared to nonsignal structures. • Sexually selected signals are more variable in size than nonsignal structures. • Measures of static scaling and trait variation offer insight into trait function. • Steep scaling and high trait variation suggest signal function in extinct taxa. [ABSTRACT FROM AUTHOR]