A growing body of research has shown that static stance control (e.g., body sway) is influenced by cognitive demands (CD), an effect that may be related to competition for limited central resources. Measures of stance control have also been impacted by postural demands (PD) (e.g., stable vs. unstable stances). However, less is known of any possible interactions between PD and CD on static stance control in populations with intact balance control and ample cognitive resources, like young healthy adults. In this study, among the same participants, we factorially compared the impact of PD with and without CD on static stance control. Thirty-four healthy young adults wore inertial measurement units (IMU) while completing static stance tasks for 30 seconds in three different PD positions: feet apart, feet together, and tandem feet. After completing these tasks alone, participants performed these tasks with CD by concurrently completing verbal serial seven subtractions from a randomly selected three-digit number. For two dependent measures, path length and jerk, there were main effects of CD and PD but no interaction effect between these factors. For all other stance control parameters, there was only a PD main effect. Thus, adding a cognitive demand to postural demands, while standing upright, may have an independent impact on stance control, but CD does not seem to interact with PD. These results suggest that young healthy adults may be less sensitive to simple PD and CD due to their greater inherent balance control and available cognitive resources. Future work might explore more complex PD and CD combinations to determine the boundaries under which young adults’ resources are taxed.