In this work, we introduce a novel application of the adaptive mesh refinement (AMR) technique for the global stability analysis of incompressible flows. The design of an accurate mesh for transitional flows is crucial since an inadequate resolution might introduce numerical noise that triggers premature transition to turbulence. With AMR, we enable the design of three different and independent meshes for the non-linear base flow and the linear direct and adjoint solutions. Each of these is designed to reduce the truncation and quadrature errors for its respective solution, based on the spectral error indicator. We provide details about the workflow and the refining procedure. The numerical framework is validated for the two-dimensional flow past a circular cylinder, computing a portion of the spectrum for the linearised direct and adjoint Navier–Stokes operators.