We analyze solutions starting from singular initial conditions in shell models of turbulence. Such initial conditions may result from a finite-time blowup, developed turbulent states, or unstable discontinuities. First, we consider an example with nonunique solutions, which are all physically relevant: An infinite number of solutions arise depending on a way the viscosity approaches zero. Next, we argue that despite of the nonuniqueness of specific physical realizations, a probability distribution for the whole set of possible solutions is unique, i.e., there is a unique spontaneously stochastic solution. This uniqueness is explained as the ordinary deterministic chaos developing in a renormalized system. The results are fully supported by numerical simulations. If time permits, I will show how these ideas can be applied in the inviscid limit of the Rayleigh-Taylor instability.