In this article we show the global multiplicity result for the following nonlocal singular problem where ω is a bounded domain in ℝn with smooth boundary ∂ω, n > 2s, s ε (0; 1), λ > 0, q > 0 satisfies q(2s - 1) < (2s + 1) and 2* s = 2n/(n - 2s). Employing the variational method, we show the existence of at least two distinct weak positive solutions for (Pλ) in X0 when λ ε (0, Λ) and no solution when λ > Λ, where Λ > 0 is appropriately chosen. We also prove a result of independent interest that any weak solution to (Pλ) is in Cα(ℝn) with α = α(s, q) ε (0, 1). The asymptotic behaviour of weak solutions reveals that this result is sharp. © 2019 Juliusz Schauder Centre for Nonlinear Studies Nicolaus Copernicus University in Toruń.