We revisit the classical theory of indentation for very soft materials. Many experiments consist in extracting the stiffness of a membrane from the cubic answer of the force versus indentation depth. However, this law is restricted to a perfect membrane under a sharp point loading. In biophysical experiments, where this technique recovers some success at low scales thanks to AFM, the thin samples are highly deformable, hyper-elastic, pre-stretched and often attached to a substrate which cannot be neglected. In addition microscopic tiny pores may exist or may be created by the indenter. This diversity requires specific studies with the correct elasticity: here we choose the Neo-Hookean elastic model at large membrane deformations. We show that, the weak loading regime is extremely sensitive to any physical properties of the thin layer but also of the indenter geometry. In addition, when a hole exists, at finite forcing or finite indentation depth, we discover a topological bifurcation with abrupt dynamical jump variation of typical quantities such as the hole size. This bifurcation is similar to the famous catenoid instabilities which are also due to a topological bifurcation, when the two rings of support are pulled apart. This bifurcation is robust in the sense that it always exists whatever the physical properties of the sample.