Some nonlinear eigenvalue problems related to the modelling of the steady-state deflection of an elastic membrane associated with a Micro-Electromechanical System capacitor under a constant applied voltage are analysed using formal asymptotic methods. These problems consist of certain singular perturbations of the basic membrane nonlinear eigenvalue problem Δu = λ/(1 + u)2 in Ω with u = 0 on ∂Ω, where Ω is the unit ball in 2. It is well known that the radially symmetric solution branch to this basic membrane problem has an infinite fold-point structure with λ → 4/9 as ϵ ≡ 1 − ||u||∞ → 0+. One focus of this paper is to develop a novel singular perturbation method to analytically determine the limiting asymptotic behaviour of this infinite fold-point structure in terms of two constants that must be computed numerically. This theory is then extended to certain generalisations of the basic membrane problem in the N-dimensional unit ball. The second main focus of this paper is to analyse the effect of two distinct perturbations of the basic membrane problem in the unit disk resulting from either a bending energy term of the form −δΔ2u to the operator, or inserting a concentric inner undeflected disk of radius δ. For each of these perturbed problems, it is numerically shown that the infinite fold-point structure for the basic membrane problem is destroyed when δ > 0, and that there is a maximal solution branch for which λ → 0 as ϵ ≡ 1 − ||u||∞ → 0+. For δ > 0, a novel singular perturbation analysis is used in the limit ϵ → 0+ to construct the limiting asymptotic behaviour of the maximal solution branch for the biharmonic problem in the unit slab and the unit disk, and for the annulus problem in the unit disk. The asymptotic results for the bifurcation curves are shown to compare very favourably with full numerical results.