In this paper we consider the spectrum and stability properties of small-amplitude waves in three-dimensional inviscid compressible swirling flow with non-zero mean vorticity, contained in an infinitely long annular circular cylinder. The mean flow has swirl and sheared axial components which are general functions of radius. We describe the form of the spectrum, in particular the three distinct types of disturbance: sonic (or acoustic) modes; nearly-convected modes; and the non-modal continuous spectrum. The phenomenon of accumulation of infinitely many eigenvalues of the nearly-convected type in the complex wavenumber-plane is classified carefully: we find two different regimes of accumulating neutral modes and one regime of accumulating instability modes, and analytic conditions for the occurrence of each type of behaviour are given. We also discuss the Green's function for the unsteady field, and in particular the contribution made by the continuous spectrum. We show that this contribution can grow algebraically downstream, and is responsible for a new type of convective instability. The algebraic growth rate of this instability is a complicated function of the mean flow parameters, and can be arbitrarily large as a function of radius in cases in which the local convected wavenumber has a local extremum. The algebraic instability we describe is additional to any conventional modal instability which may be present, and indeed we exhibit cases which are convectively stable to modes, but which nevertheless grow algebraically downstream.