Handedness is a human behavioural phenotype that appears to be congenital, and is often assumed to be inherited, but for which the developmental origin and underlying causation(s) have been elusive. Models of the genetic basis of variation in handedness have been proposed that fit different features of the observed resemblance between relatives, but none has been decisively tested or a corresponding causative locus identified. In this study, we applied data from well-characterised individuals studied at the London Twin Research Unit. Analysis of genome-wide SNP data from 3940 twins failed to identify any locus associated with handedness at a genome-wide level of significance. The most straightforward interpretation of our analyses is that they exclude the simplest formulations of the 'right-shift' model of Annett and the 'dextral/chance' model of McManus, although more complex modifications of those models are still compatible with our observations. For polygenic effects, our study is inadequately powered to reliably detect alleles with effect sizes corresponding to an odds ratio of 1.2, but should have good power to detect effects at an odds ratio of 2 or more.