Controlled laboratory experiments reveal that the lower part of turbidity currents has the ability to enter fluid mud substrates, if the bed shear stress is higher than the yield stress of the fluid mud and the density of the turbidity current is higher than the density of the substrate. Upon entering the substrate, the turbidity current either induces mixing between flow-derived sediment and substrate sediment, or it forms a stable horizontal flow front inside the fluid mud. Such ‘intrabed’ flow is surrounded by plastically deformed mud; otherwise it resembles the front of a ‘bottom-hugging’ turbidity current. The ‘suprabed’ portion of the turbidity current, i.e. the upper part of the flow that does not enter the substrate, is typically separated from the intrabed flow by a long horizontal layer of mud which originates from the mud that is swept over the top of the intrabed flow and then incorporated into the flow. The intrabed flow and the mixing mechanism are specific types of interaction between turbidity currents and muddy substrates that are part of a larger group of interactions, which also include bypass, deposition, erosion and soft sediment deformation. A classification scheme for these types of interactions is proposed, based on an excess bed shear stress parameter, which includes the difference in the bed shear stress imposed by the flow and the yield stress of the substrate and an excess density parameter, which relies on the density difference between the flow and the substrate. Based on this classification scheme, as well as on the sedimentological properties of the laboratory deposits, an existing facies model for intrabed turbidites is extended to the other types of interaction involving soft muddy substrates. The physical threshold of flow-substrate mixing versus stable intrabed flow is defined using the gradient Richardson number, and this method is successfully validated with the laboratory data. The gradient Richardson number is also used to verify that stable intrabed flow is possible in natural turbidity currents, and to determine under which conditions intrabed flow is likely to be unstable. It appears that intrabed flow is likely only in natural turbidity currents with flow velocities well below c. 3.5 m s−1, despite the fact that a wider range of flows is capable of entering fluid muds. Below this threshold velocity, intrabed flow is stable only at high density gradients and low velocity gradients across the upper boundary of the turbidity current. Finally, the gradient Richardson number is used as a scaling parameter to set the flow velocity limits of a natural turbidity current that formed an inferred intrabed turbidite in the deep-marine Aberystwyth Grits Group, West Wales, United Kingdom.