Self-driving cars enable dynamic shared mobility, where customers are independent of schedules and fixed stops. This study aims to investigate the potential effects shared mobility can have on future transportation. We simulate multiple scenarios to analyze the effects different service designs might have on vehicle kilometers, on the required number of shared vehicles, on the potential replacement of private cars, and on service metrics such as waiting times, travel times, and detour levels. To demonstrate how simulation can be used to analyze future mobility, we present a case study of the city of Gothenburg in Sweden, where we model travel demand in the morning hours of a workday. The results show that a significant decrease of vehicle kilometers can be achieved if all private car trips are replaced by rideshare and that shared vehicles can potentially replace at least 5 private cars during the morning peak.