Synaesthesia is a neurological phenomenon affecting perception, where triggering stimuli (e.g. letters and numbers) elicit unusual secondary sensory experiences (e.g. colours). Family-based studies point to a role for genetic factors in the development of this trait. However, the contributions of common genomic variation to synaesthesia have not yet been investigated. Here, we present the SynGenes cohort, the largest genotyped collection of unrelated people with grapheme–colour synaesthesia (n = 723). Synaesthesia has been associated with a range of other neuropsychological traits, including enhanced memory and mental imagery, as well as greater sensory sensitivity. Motivated by the prior literature on putative trait overlaps, we investigated polygenic scores derived from published genome-wide scans of schizophrenia and autism spectrum disorder (ASD), comparing our SynGenes cohort to 2181 non-synaesthetic controls. We found a very slight association between schizophrenia polygenic scores and synaesthesia (Nagelkerke's R2 = 0.0047, empirical p = 0.0027) and no significant association for scores related to ASD (Nagelkerke's R2 = 0.00092, empirical p = 0.54) or body mass index (R2 = 0.00058, empirical p = 0.60), included as a negative control. As sample sizes for studying common genomic variation continue to increase, genetic investigations of the kind reported here may yield novel insights into the shared biology between synaesthesia and other traits, to complement findings from neuropsychology and brain imaging.