Simulators are widely used to test Autonomous Driving Systems (ADS), but their potential flakiness can lead to inconsistent test results. We investigate test flakiness in simulation-based testing of ADS by addressing two key questions: (1) How do flaky ADS simulations impact automated testing that relies on randomized algorithms? and (2) Can machine learning (ML) effectively identify flaky ADS tests while decreasing the required number of test reruns? Our empirical results, obtained from two widely-used open-source ADS simulators and five diverse ADS test setups, show that test flakiness in ADS is a common occurrence and can significantly impact the test results obtained by randomized algorithms. Further, our ML classifiers effectively identify flaky ADS tests using only a single test run, achieving F1-scores of 85%, 82% and 96% for three different ADS test setups. Our classifiers significantly outperform our non-ML baseline, which requires executing tests at least twice, by 31%, 21%, and 13% in F1-score performance, respectively. We conclude with a discussion on the scope, implications and limitations of our study. We provide our complete replication package in a Github repository (Github paper 2023).