Are crumpled sheets marginally stable?

We study networks of coupled bistable elastic elements, recently proposed as a model for crumpled thin sheets. The networks are poised on the verge of a localized instability, and the model allows unique access to both local and global properties associated with marginal stability. We directly measure pseudo-gaps in the spectrum of local excitations, as well as diverging fluctuations under shear. The networks also host quasi-localized, low-frequency vibrational modes, and scale-free avalanches of instabilities. We propose a correction to the scaling between the pseudo-gap exponent and avalanche statistics based on diverging length fluctuations. Crucially, the dynamics are dominated by a small population of bonds which are locally unstable. Our model combines a coarse-grained view with a continuous, real-space implementation, providing novel insights to a wide class of amorphous solids.