The extracellular matrix (ECM) is a highly organised structure with the capacity to direct cell migration through their tendency to follow matrix fibres, a process known as contact guidance. Amoeboid cell populations migrate in the ECM by making frequent shape changes and have minimal impact on its structure. Mesenchymal cells actively remodel the matrix to generate the space in which they can move. In this paper, these different types of movement are studied through simulation of a continuous transport model. It is shown that the process of contact guidance in a structured ECM can spatially organise cell populations. Furthermore, when combined with ECM remodelling, it can give rise to cellular pattern formation in the form of "cell-chains" or networks without additional environmental cues such as chemoattractants. These results are applied to a simple model for tumour invasion where it is shown that the interactions between invading cells and the ECM structure surrounding the tumour can have a profound impact on the pattern and rate of cell infiltration, including the formation of characteristic "fingering" patterns. The results are further discussed in the context of a variety of relevant processes during embryonic and adult stages.