Rotating Detonation Combustor (RDC) operation is investigated under different air and fuel flow rates, and varied geometries to reveal four fundamentally different instabilities. Select test points are chosen to study these instabilities using qualitative and quantitative tools. First, for RDC operation at lean limits, or with subsonic air injection, or with large fuel injection orifices, the detonation wave inside the combustor undergoes aperiodic chaotic propagation around the combustor annulus characterized by incoherent pressure-time traces. The high incoherence in recorded pressure, along with the considerable variation in subsequent pressure peaks suggests numerous failure and re-initiation of the detonation wave. Second, almost all operating points at the variety of conditions tested exhibit some degree of low frequency sinusoidal oscillations characterized by periodic waxing and waning of subsequent detonation peak pressures. It occurs between 200 Hz and 500 Hz for the operating maps studied. Third, the phenomenon of mode switching in the RDC is also defined as instability since the sudden change in the number of detonation waves existing in the combustor is temporally unpredictable, and often unpredictable for a given geometry. It is found that RDC operation is more stable when there are multiple detonation waves inside the chamber. With a back-pressurizing convergent nozzle, at certain operating points, the RDC exhibits axisymmetric pulsed operation like the Pulsed Detonation Combustor (PDC). There is significant evidence to suggest that these longitudinal pulsed detonations (LPD) are manifested due to shock-reflection from the RDC exit followed by a subsequent shock-initiation of the fresh reactants. The frequency of this instability is around 3.8 kHz for the test case investigated. Copyright (C) 2015, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.