We present an alternative form of intermittency, Lévy on-off intermittency, which arises from multiplicative α-stable white noise close to an instability threshold. We study this problem in the linear and nonlinear regimes, both theoretically and numerically, for the case of a pitchfork bifurcation with fluctuating growth rate. We compute the stationary distribution analytically and numerically from the associated fractional Fokker-Planck equation in the Stratonovich interpretation. We characterize the system in the parameter space (α,β) of the noise, with stability parameter α∈(0,2) and skewness parameter β∈[−1,1]. Five regimes are identified in this parameter space, in addition to the well-studied Gaussian case α=2. Three regimes are located at 1<α<2, where the noise has finite mean but infinite variance. They are differentiated by β and all display a critical transition at the deterministic instability threshold, with on-off intermittency close to onset. Critical exponents are computed from the stationary distribution. Each regime is characterized by a specific form of the density and specific critical exponents, which differ starkly from the Gaussian case. A finite or infinite number of integer-order moments may converge, depending on parameters. Two more regimes are found at 0<α≤1. There, the mean of the noise diverges, and no critical transition occurs. In one case, the origin is always unstable, independently of the distance μ from the deterministic threshold. In the other case, the origin is conversely always stable, independently of μ. We thus demonstrate that an instability subject to nonequilibrium, power-law-distributed fluctuations can display substantially different properties than for Gaussian thermal fluctuations, in terms of statistics and critical behavior.