In this computational study, stabilized finite element solutions of convection-dominated stationary and linear reaction–convection–diffusion equations are studied. Although the standard (Bubnov–) Galerkin finite element method (GFEM) is one of the most robust, efficient, and reliable methods for solving many engineering problems and scientific computations, it typically suffers from numerical instabilities when solving convection-dominated problems. Towards that end, this work deals with a stabilized version of the standard GFEM, called the streamline-upwind/Petrov–Galerkin (SUPG) formulation, to overcome the instability issues arising when solving such problems. The SUPG-stabilized formulation is further supplemented with a shock-capturing mechanism, called YZ β shock-capturing, to provide additional stability around steep gradients. A comprehensive set of test examples is provided to assess and compare the performances of the proposed methods, i.e., the GFEM, SUPG, and SUPG-YZ β . It is observed that the GFEM approximations involve spurious oscillations for small values of the diffusion parameter, as expected. We observe through numerical experiments that such globally spread oscillations are suppressed significantly when the SUPG formulation is employed. It is also demonstrated that the SUPG-stabilized formulation needs additional stability to resolve localized sharp layers, and the SUPG-YZ β formulation yields better shock resolutions for such regions.