This work compares the numerical performance of the virtual element method (VEM) with a fully discretized scaled boundary finite element method (SBFEM) for crack propagation problems. Both approaches include higher-order discretization for polygonal meshes, which have proven to be convenient when modelling cracked structures and provide superior flexibility in mesh design compared to conventional finite element methods (FEM). The VEM formalism is based on employing polynomial projections of the displacement field, allowing for arbitrary element shapes to be utilized in the discretizations. This enables straightforward integration of hanging nodes and thus facilitates adaptive mesh modification without remeshing. These features prove particularly useful in the context of discrete crack growth simulations, as controlling the element quality in the vicinity of the moving crack tip is essential for accurately calculating the crack loading, based on the theory of configurational forces or the J-integral. In comparison to VEM, the SBFEM commonly refers to stress intensity factors (SIFs) which are used for the determination of the crack propagation angle. These can be determined with the relative crack mouth displacements, thereby circumventing cumbersome post-processing steps. The singularity of the near-field stresses is naturally included in the stress definition of open SBFEM crack tip elements. However, the eigenvalue problem needs to be solved in each crack step and the transfer to nonlinear problems can be challenging. Therefore, several authors constructed interpolation functions based on SBFEM equations, resulting in an efficient approximation of the displacement field for polygonal shapes that can easily be incorporated in hybrid meshes. The aim of this work is the efficient simulation of crack trajectories, exploiting and comparing the respective advantages of the aforementioned polygonal methods. Benchmarks are performed for problems of linear elastic fracture mechanics, accounting for mixed-mode loading as well as for contact interface constraints.