Nowadays, the reliability has become one of the main issues for safety-critical embedded systems, like automotive, aerospace and avionic. In an embedded system, the full system stack usually includes, between the hardware layer and the software/application layer, a middle layer composed by the Operating System (OS) and the middleware. Most of the time, in the literature only the application-layer is considered during the reliability analysis. This is due to the fact that middle layer short execution time makes the probability of a fault affecting it much lower compared to the application-level. Nevertheless, middle layer data structures lifespan is equivalent to the application layer ones. Moreover, all the times a hardware fault propagates to the middle-layer as an error, and especially to the OS, its impact can be expected to be potentially catastrophic. The aim of this work is to study the reliability of a Real-Time Operating System (RTOS) affected by Single Event Upset (SEU) faults. The methodology targets the most relevant variables and data structures of FreeRTOS analyzed through a software-based fault injection. Results show the ability to highlight the criticality in the OS fault tolerance, in terms of system integrity, data integrity and the overall inherent resiliency to faults, potentially leading to selective hardening of the OS.