The existence of undulations of the geoid, gravity and bathymetry in ocean basins, as well as anomalies in heat flow, point to the existence of small scale convection beneath tectonic plates. The instabilities that could develop at the base of the lithosphere are sufficiently small scale (\textless500 km) that they remain mostly elusive from seismic detection. We take advantage of 3D spherical numerical geodynamic models displaying plate-like behavior to study the interaction between large-scale flow and small-scale convection. We find that finger-shaped instabilities develop at seafloor ages \textgreater60 Ma. They form networks that are shaped by the plate evolution, slabs, plumes and the geometry of continental boundaries. Plumes impacting the boundary layer from below have a particular influence through rejuvenating the thermal lithosphere. They create a wake in which new instabilities form downstream. These wakes form channels that are about 1000 km wide, and thus are possibly detectable by seismic tomography. Beneath fast plates, cold sinking instabilities are tilted in the direction opposite to plate motion, while they sink vertically for slow plates. These instabilities are too small to be detected by usual seismic methods, since they are about 200 km in lateral scale. However, this preferred orientation of instabilities below fast plates could produce a pattern of large-scale azimuthal anisotropy consistent with both plate motions and the large scale organisation of azimuthal anisotropy obtained from recent surface wave models. (C) 2018 Elsevier B.V. All rights reserved.