We present a stochastic and spatial Monte Carlo model for the growth of a fungal colony in microstructures. This model is based on an 'L-system-like' representation of filaments as individual objects. Each of these can both grow in space (and be diverted by obstacles) and can send new branches. All parameters in the model such as filament dimensions, the growth speed, behavior at and around obstacles, branching angle and frequency and others are obtained from experimental studies of growth in artificial microstructures. We investigate four different possible 'strategies; the colony might use to achieve the tasks of (a) filling the available space and (2) finding its way out of the structures. The simulation results indicate that a combination of directional memory and a stop-and-branch behavior at corners gives the best results and observe that in fact this is similar to the experimentally observed behavior of the fungi. The model is expected to be of use in studying the colonization of microstructures by fungi and in the design of devices either using fungal growth or aiming to inhibit it.