In the junction Ω of several semi-infinite cylindrical waveguides, the Dirichlet Laplacian is treated whose continuous spectrum is the ray [λ_†,+∞) with a positive cutoff value λ_†. Two different criteria are presented for the threshold resonance generated by nontrivial bounded solutions to the Dirichlet problem for the Helmholtz equation −Δu = λ_†u in Ω. The first criterion is quite simple andis convenient to disprove the existence of bounded solutions. The second criterion is rather involved but can help to detect concrete shapes supporting the resonance. Moreover, the latter distinguishes in a natural way between stabilizing, i.e., bounded but nondecaying solutions, and trapped modes with exponential decay at infinity