Unlike macroscopic multistable mechanical systems such as snap bracelets or elastic shells that must
be physically manipulated into various conformations, microscopic systems can undergo spontaneous
conformation switching between multistable states due to thermal fluctuations. Here we investigate the
statistical mechanics of shape transitions in small elastic elliptical plates and shells driven by noise. By
assuming that the effects of edges are small, which we justify exactly for plates and shells with a lenticular
section, we decompose the shapes into a few geometric modes whose dynamics are easy to follow. We use
Monte Carlo simulations to characterize the shape transitions between conformational minimal as a
function of noise strength, and corroborate our results using a Fokker-Planck formalism to study the
stationary distribution and the mean first passage time problem. Our results are applicable to objects such as
graphene flakes or protein β sheets, where fluctuations, geometry, and finite size effects are important.