This Letter addresses the dynamical quantum problem of a driven discrete energy level coupled to a semi-infinite continuum whose density of states has a square-root-type singularity, such as states of a free particle in one dimension or quasiparticle states in a BCS superconductor. The system dynamics is strongly affected by the quantum-mechanical repulsion between the discrete level and the singularity, which gives rise to a bound state, suppresses the decay into the continuum, and can produce Stueckelberg oscillations. This quantum coherence effect may limit the performance of mesoscopic superconducting devices, such as the quantum electron turnstile.