In this article, we show the global multiplicity result for the following nonlocal singular problem (P λ) : (−∆) s u = u −q + λu 2 * s −1 , u > 0 in Ω, u = 0 in R n \ Ω, where Ω is a bounded domain in R n with smooth boundary ∂Ω, n > 2s, s ∈ (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 (P λ) in X 0 when λ ∈ (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 (P λ) is in C α (R n) with α = α(s, q) ∈ (0, 1). The asymptotic behaviour of weak solutions reveals that this result is sharp.