Fault detection and exclusion (FDE) is the main task for pre-processing of global navigation satellite system (GNSS) positions and is a fundamental process in integrity monitoring that is needed to achieve reliable positioning for applications such as in intelligent transport systems. A widely used method is the solution separation (SS) algorithm. The FDE in SS traditionally builds the models assuming positioning errors are normally distributed. However, in urban environments, this traditional assumption may no longer be valid. The objective of this study is to investigate this and further examine the performance of alternative distributions, which can be useful for FDE modelling and thus improved navigation. In particular, it investigates characterization of positioning errors using GNSS when the Australian satellite-based augmentation system (SBAS) test bed is used, which comprised different positioning modes, including single-point positioning (SPP) using the L1 global positioning system (GPS) legacy SBAS, the second-generation dual-frequency multi-constellation (DFMC) SBAS service for GPS and Galileo, and, finally, precise point positioning (PPP) using GPS and Galileo observations. Statistical analyses are carried out to study the position error distributions over different possible operational environments, including open sky, low-density urban environment, and high-density urban environment. Significant autocorrelation values are also found over all areas. This, however, is more evident for PPP solution. Furthermore, the applied distribution analyses applied suggest that in addition to the normal distribution, logistic, Weibull, and gamma distribution functions can fit the error data in various cases. This information can be used in building more representative FDE models according to the work environment.