The lake equations arise as a geophysical model for the description of shallow water. The system is introduced as a 2D model for the vertically averaged horizontal component of a 3D incompressible fluid. A lake is characterised by a 2D domain and a non-negative topography function. The 2D velocity satisfies an anelastic constraint rather than a divergence-free condition. The equations are degenerate if the topography may vanish. More precisely, velocity and vorticity are then related through degenerate elliptic problems. In this talk, we discuss the stability of the lake equations for singular geometries and degenerated topographies. Specifically, we prove stability results for two scenarios: First, motivated by natural phenomena such as flooding or erosion we consider a sequence of lakes with an island that disappears. In addition, we highlight crucial differences to the incompressible 2D Euler equations (flat topography). Second, we address the stability of the equations for a sequence of lakes for which an island appears in the limit, e.g. due to a decreasing level of water. This is joint work with C. Lacave and E. Miot.