We consider meniscus instabilities in thin elastic layers perfectly adhered to, and confined between, much stiffer bodies. When the free boundary associated with the meniscus of the elastic layer recedes into the layer, for example by pulling the stiffer bodies apart or injecting air between them, then the meniscus will eventually undergo a purely elastic instability in which fingers of air invade the layer. Here we show that the form of this instability is identical in a range of different loading conditions, provided only that the thickness of the meniscus, a, is small compared to the in-plane dimensions and to two emergent in-plane length scales that arise if the substrate is soft or if the layer is compressible. In all such situations, we predict that the instability will occur when the meniscus has receded by approximately 1.27a, and that the instability will have wavelength λ ≈ 2.75a. We illustrate this by also calculating the threshold for fingering in a thin wedge of elastic material bonded to two rigid plates that are pried apart, and the threshold for fingering when a flexible plate is peeled from an elastic layer that glues the plate to a rigid substrate.