Polygenic scores (PGS) are commonly evaluated in terms of their predictive accuracy at the population level by the proportion of phenotypic variance they explain. To be useful for precision medicine applications, they also need to be evaluated at the individual patient level when phenotypes are not necessarily already known. Hence, we investigated the stability of PGS in European-American (EUR)- and African-American (AFR)-ancestry individuals from the Philadelphia Neurodevelopmental Cohort (PNC) and the Adolescent Brain Cognitive Development Study (ABCD) using different discovery GWAS for post-traumatic stress disorder (PTSD), type-2 diabetes (T2D), and height. We found that pairs of EUR-ancestry GWAS for the same trait had genetic correlations > 0.92. However, PGS calculated from pairs of same-ancestry and different-ancestry GWAS had correlations that ranged from <0.01 to 0.74. PGS stability was higher for GWAS that explained more of the trait variance, with height PGS being more stable than PTSD or T2D PGS. Focusing on the upper end of the PGS distribution, different discovery GWAS do not consistently identify the same individuals in the upper quantiles, with the best case being 60% of individuals above the 80th percentile of PGS overlapping from one height GWAS to another. The degree of overlap decreases sharply as higher quantiles, less heritable traits, and different-ancestry GWAS are considered. PGS computed from different discovery GWAS have only modest correlation at the level of the individual patient, underscoring the need to proceed cautiously with integrating PGS into precision medicine applications.