We establish several results related to existence, nonexistence or bifurcation of positive solutions for the boundary value problem −Δu+K(x)g(u)+|∇u|a=λf(x,u) in Ω, u=0 on ∂Ω, where Ω⊂RN (N⩾2) is a smooth bounded domain, 0<a⩽2, λ is a positive parameter, and f is smooth and has a sublinear growth. The main feature of this paper consists in the presence of the singular nonlinearity g combined with the convection term |∇u|a. Our approach takes into account both the sign of the potential K and the decay rate around the origin of the singular nonlinearity g. The proofs are based on various techniques related to the maximum principle for elliptic equations.