At the continuum level, we study the pattern selection mechanism of finger-like crack growth phenomena in gradient driven growth problems in general, and the structure of stress-induced morphological instabilities in crazing of polymer glasses in particular. We simulate solidification in a narrow channel through the use of a phase-field model with an adaptive grid. By tuning a dimensionless parameter, the Peclet number, we show a continuous crossover from a free dendrite at high Peclet numbers to anisotropic viscous fingering at low Peclet numbers. At low Peclet numbers we find good agreement between our results, theoretical predictions, and experiment, providing the first quantitative test of solvability theory for anisotropic viscous fingers. For high undercoolings, we find new phenomena, a solid forger which satisfies stability and thermodynamic criterion. We further provide an analytical form for the shape of these fingers, based on local models of solidification, which fits our numerical results from simulation. Later we study the growth of crazes in polymer glasses by deriving the equations of motion of plastic flow at the craze tip, and the steady-state velocity profile of this flow. By developing a phenomenological model, we solve the full time-dependent equations of motion of this highly non-linear phenomena. Our simulation produces the steady-state cellular pattern observed in experiments. We further show that polymer glasses with lower yield stress produce cellular patterns with sharper tips and more cells, indicating instabilities with smaller wavelengths.