Irregular scattering at harmonically driven one-dimensional potential wells is studied both on the classical and the quantum level. We show that an ac-driven single square well, and a smooth well with oscillating bottom, are sufficient to generate chaotic scattering. For a square well with oscillating bottom, we introduce the concept of pseudointegrable scattering. The quantum dynamics of these models is treated using Floquet scattering theory, which is exact for arbitrary amplitude and frequency of the driving. In the deep quantum regime, scattering is dominated by multiphoton exchanges with the driving field, leading to complex resonance structures in transmission and reflection. For strong and fast driving, the ac-driven square well develops an effective double-well potential that introduces coherent tunneling in the scattering. We identify signatures of classical chaotic scattering in a phase-space representation of the quantum dynamics.