In the finite element modelling of masonry structures, the micro-modelling technique of differentiating the continuum into a linear elastic bulk, and interfaces representing non-linear joints is common. However, this approach of simulating cracking-crushing-shearing failure possibilities in interfaces, typical of damage in masonry, also poses numerical stability issues due to the quasi-brittle nature of the failure. In this regard, the article proposes the use of numerically robust sequentially linear procedures and a suitable discretised tension-shear-compression failure model for interfaces. Sequentially linear solution procedures describe the nonlinear response of a specimen/structure through a sequence of scaled linear analyses, each of which represents locally applied damage increments, using secant-stiffness based discretised constitutive relations called saw-tooth laws. The constitutive formulation proposed herein includes a tension cut-off criterion combined with a uniaxial discretised softening law, a Coulomb friction criterion with a discretised cohesion softening law, and a compression cut-off criterion combined with a uniaxial discretised hardening–softening law. It is presented for both two-dimensional (2D) line interfaces and three-dimensional (3D) planar interfaces. The applicability of these formulations are illustrated using 2D and 3D models of a pushover analysis on a squat unreinforced masonry wall. The simulations are made using Sequentially Linear Analysis (SLA) and the Force-Release method, which are total (load-unload) and incremental sequentially linear methods respectively. The clear global softening in the force–displacement evolution and the localised brittle shear failure observed in the experiment are reproduced well and in a stable manner.