Based on the estimates of accuracy, genomic selection would be useful for selecting for improved trait values and trait stability for agronomic and quality traits in wheat. Trait values and trait stability estimated by two methods were generally independent indicating a breeder could select for both simultaneously. Genomic selection (GS) is a new marker-assisted selection tool for breeders to achieve higher genetic gain faster and cheaper. Breeders face challenges posed by genotype by environment interaction (GEI) pattern and selecting for trait stability. Obtaining trait stability is costly, as it requires data from multiple environments. There are few studies that evaluate the efficacy of GS for predicting trait stability. A soft winter wheat population of 273 lines was genotyped with 90 K single nucleotide polymorphism markers and phenotyped for four agronomic and seven quality traits. Additive main effect and multiplicative interaction (AMMI) model and Eberhart and Russell regression (ERR) were used to estimate trait stability. Significant GEI variation was observed and stable lines were identified for all traits in this study. The accuracy of GS ranged from 0.33 to 0.67 for most traits and trait stability. Accuracy of trait stability was greater than trait itself for yield (0.44 using AMMI versus 0.33) and heading date (0.65 using ERR versus 0.56). The opposite trend was observed for the other traits. GS did not predict the stability of the quality traits except for flour protein, lactic acid and softness equivalent. Significant GS accuracy for some trait stability indicated that stability was under genetic control for these traits. The magnitude of GS accuracies for all the traits and most of the trait stability index suggests the possibility of rapid selection for these trait and trait stability in wheat breeding.