As we witness a global change occurring with the advent of e-scooters (electric scooters), it is clear that adequate measures need to be taken for the implementation of this means of transport. Bearing in mind that the user should be the focus of the proposed measures, the aim of this paper encompasses the analysis of e-scooter users’ travel patterns, as well as standard scooter users’ willingness to switch to e-scooters for different hypothetical scenarios, and the determination of factors that have an influence on the users’ willingness to switch. The scenarios include the availability of various infrastructure capacities for the city of Belgrade, namely, a separate infrastructure for e-scooters, or sharing the existing infrastructure with cyclists, pedestrians and motor vehicles. Standard methods of descriptive statistics, the McNemar–Bowker test of paired samples, and multinomial logistic regression were used in this paper. The results showed that there was a statistically significant difference in users’ willingness to use an e-scooter, depending on the available infrastructure. The results of multinomial logistic regression showed that the mode of transport used before and during the COVID-19 pandemic, as well as the average distance traveled, had a statistically significant impact on the users’ willingness to switch to e-scooters if they were allowed to use the cycling infrastructure. This paper also identifies additional factors that have a positive (environmental benefits, congestion avoidance) and negative (safety issues, lack of infrastructure, etc.) impact on users’ willingness to switch, which can be of use to decision-makers as a basic guideline for the adequate implementation of e-scooters in transport systems.