Generation-IV international forum has identified thermal-hydraulic (TH) stability and safety issues as important challenges in the operation of supercritical water reactor (SCWR). In the present work, the scheme of the U. S. SCWR design is analyzed numerically for the TH static and dynamic stability, and safety concerns by employing an in-house TH model which solves nonlinearly coupled mass, axial momentum and energy conservation equations along with the pressure and specific enthalpy dependent thermodynamic equation of state using a characteristics-based implicit time-domain solution methodology to account for the compressibility of the supercritical water (SCW) dynamics. The applicability of the model for the aforementioned purposes has been verified against the state-of-the-art available results. Next, the model is used for the analysis of static instability due to Ledinegg excursion for the U. S. SCWR and results predict the possibility of the instability for the reactor is only at hypothetical operating conditions. The analysis of dynamic instability due to density wave oscillations (DWOs) is performed next. The investigations find its possibility in the relevant operating regime and the stability thresholds are identified to obtain marginal stability boundary (MSBs) for the reactor. Parametric studies are carried out next to determine the effect of exit pressure and coolant mass flow rate on the MSB. Further, the zone of safe operation from metallurgical point of view for the U. S. SCWR at normal and deteriorated heat transfer (DHT) regions is identified after determining the axial variation of heat transfer rate, clad wall temperature and various DHT parameters. Finally, a common stable and safe zone is obtained.