We study the stick-slip instability when peeling an adhesive tape from a flat substrate at an imposed driving velocity. Using a high-speed camera mounted on a microscope, we image either the detachment front, or the ribbon profile at microscopic scale, while simultaneously recording acoustic emissions. In a given range of driving velocity, the peeling displays a multi-scale unstable dynamics, with two well-separated spatio-temporal scales, which corresponds to microscopic and macroscopic dynamical stick-slip instabilities. We show that the microscopic stick-slip dynamics -- which presents very different characteristics than the well-known macroscopic instability -- is related to a high-frequency periodic release of the elastic bending energy of the adhesive ribbon concentrated at the vicinity of the peeling front. Moreover, we also show that it corresponds to the recently observed periodic propagation of fast transverse fractures across the tape width in the ultrasonic range.