In this paper, we investigate the singular Sturm-Liouville problem u′′=λg(u), u′(0)=0, βu′(1)+αu(1)=A, where λ is a nonnegative parameter, β≥0, α>0, and A>0. We discuss the existence of multiple positive solutions and show that for certain values of λ, there also exist solutions that vanish on a subinterval [0,ρ]⊂[0,1), the so-called dead core solutions. The theoretical findings are illustrated by computational experiments for g(u)=1/u and for some model problems from the class of singular differential equations (ϕ(u′))′+f(t,u′)=λg(t,u,u′) discussed in Agarwal et al. (2007). For the numerical simulation, the collocation method implemented in our MATLAB code bvpsuite has been applied.