Individuals most often use several rather than one substance among alcohol, cigarettes or cannabis. This widespread co-occurring use of multiple substances is thought to stem from a common liability that is partly genetic in origin. Genetic risk may indirectly contribute to a common liability to substance use through genetically influenced mental health vulnerabilities and individual traits. To test this possibility, we used polygenic scores indexing mental health and individual traits and examined their association with the common versus specific liabilities to substance use. We used data from the Avon Longitudinal Study of Parents and Children (N = 4218) and applied trait-state-occasion models to delineate the common and substance-specific factors based on four classes of substances (alcohol, cigarettes, cannabis and other illicit substances) assessed over time (ages 17, 20 and 22). We generated 18 polygenic scores indexing genetically influenced mental health vulnerabilities and individual traits. In multivariable regression, we then tested the independent contribution of selected polygenic scores to the common and substance-specific factors. Our results implicated several genetically influenced traits and vulnerabilities in the common liability to substance use, most notably risk taking (bstandardised = 0.14; 95% confidence interval [CI] [0.10, 0.17]), followed by extraversion (bstandardised = -0.10; 95% CI [-0.13, -0.06]), and schizophrenia risk (bstandardised = 0.06; 95% CI [0.02, 0.09]). Educational attainment (EA) and body mass index (BMI) had opposite effects on substance-specific liabilities such as cigarette use (bstandardised-EA = -0.15; 95% CI [-0.19, -0.12]; bstandardised-BMI = 0.05; 95% CI [0.02, 0.09]) and alcohol use (bstandardised-EA = 0.07; 95% CI [0.03, 0.11]; bstandardised-BMI = -0.06; 95% CI [-0.10, -0.02]). These findings point towards largely distinct sets of genetic influences on the common versus specific liabilities.