The structure of non-trivial non-negative solutions to singularly perturbed semilinear Dirichlet problems of the form −ε2Δu = f(u) in Ω, u = 0 on ∂Ω, Ω ⊂ RN a bounded smooth domain, is studied as ε → 0+, for a class of nonlinearities f(u) satisfying f(0) = f(z1) = f(z2) = 0 with 0 < z1 < z2, f < 0 in (0, z1), f > 0 in (z1, z2) and . It is shown that there are many non-trivial non-negative solutions and they are spike-layer solutions. Moreover, the measure of each spike layer is estimated as ε → 0+. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0, ∞). Uniqueness of a large positive solution and many positive intermediate spike-layer solutions are obtained for ε sufficiently small.