Genetically identical cells in the same population can take on phenotypically variable states, leading to differentiated responses to external signals, such as nutrients and drug-induced stress. Many models and experiments have focused on a description based on discrete phenotypic states. Here, we consider the effects of selection acting on a single trait, which we explicitly link to the variable number of proteins expressed by a gene. Considering different regulatory models for the gene under selection, we calculate the steady-state distribution of expression levels and show how the population adapts its expression to enhance its fitness. We quantitatively relate the overall fitness of the population to the heritability of expression levels and their diversity within the population. We show how selection can increase or decrease the variability in the population, alter the stability of bimodal states, and impact the switching rates between metastable attractors.