We investigate (i) pull-in instabilities in a microelectromechanical (MEM) beam due to the Coulomb force and in MEM membranes due to the Coulomb and the Casimir forces, and (ii) thermomechanical instability in a heat-conducting thermoelastoviscoplastic solid due to thermal softening overcoming hardening caused by strain- and strain-rate effects. Each of these nonlinear multiphysics problems is analyzed by the meshless local Petrov-Galerkin (MLPG) method. The moving least squares (MLS) approximation is used to generate basis functions for the trial solution, and the basis for test functions is taken to be either the weight functions used in the MLS approximation, or the same as for the trial solution. In this case the method becomes Bubnov-Galerkin. Essential (displacement, temperature, electric potential) boundary conditions are enforced by the method of Lagrange multipliers. For the electromechanical problem, the pull-in voltage and the corresponding deflection are extracted by combining the MLPG method with either the displacement iteration pull-in extraction algorithm or the pseudoarclength continuation method. For the thermomechanical problem, the localization of deformation into narrow regions of intense plastic deformation is delineated. For every problem studied, computed results are found to compare well with those obtained either analytically or by the finite element (FE) method. For the same accuracy, the MLPG method generally requires fewer nodes but more CPU time than the FE method;​ thus additional computational cost is compensated somewhat by the increased efficiency of the MLPG method.