In this paper, we review some aspects of viscous fingering in a Hele-Shaw cell that at first sight appear to defy intuition. These include singular effects of surface tension relative to the corresponding zero-surface-tension problem both for the steady and unsteady problem. They also include a disproportionately large influence of small effects like local inhomogeneity of the flow field near the finger tip, or of the leakage term in boundary conditions that incorporate realistic thin-film effects. Through simple explicit model problems, we demonstrate how such properties are not unexpected for a system approaching structural instability or ill-posedness.