Three series of laboratory experiments were run, varying the flume water discharge and the downstream water depth, to investigate local scouring downstream of a trapezoidal drop followed by a rigid apron. Three different kinds of jets, developing downstream of the drop, were observed: a submerged jet, a surface wave jet, and an oscillating jet. The oscillating jet is defined as a jet which changes intermittently between a surface wave jet and a submerged jet. A different development of the scouring process due to the different kinds of jet was observed: a submerged jet produced a local scouring with a downstream dune, while a surface wave jet produced a local scouring with both an upstream and a downstream dune; the scouring process due to the oscillating jet was the result of the interaction of the scouring processes due to a submerged jet and a surface wave jet. A stability analysis of the surface wave jet was performed, with both a flat and a scoured bed, by means of the modified De Saint--Venant equations, that is, with correction terms accounting for the curvature of the streamlines. The stability of the surface wave jet is affected by the presence of the scoured bed, which weakens its stability properties.