In this paper, we link numerical observations of spiral breakup to a stability analysis of simple rotating spirals. We review the phenomenology of spiral breakup, important applications in pattern formation and the state of the art in numerical stability analysis of spirals. A strategy for the latter procedure is suggested. Phenomenologically, spiral breakup can occur near the centre of rotation ('core breakup') or far away from it ('far-field breakup'). It may be accompanied by instabilities of the spiral core in particular spiral meandering that affect also the stability of waves in the far-field, because an unstable core acts as a moving source and introduces a (nonlinear) Doppler effect. In general, breakup of non-meandering spirals is related to an absolute instability of the planar wavetrain with the same wave number. To simplify the stability problem, we consider a one-dimensional cut ('1D spiral') with a fixed core position in simulations and compare the results with a stability analysis of planar wavetrains. These 1D spirals approximate the radial dynamics of non-meandering 2D spirals. To fully account for instabilities of 1D spirals, it is not sufficient to compute the direction of propagation of the unstable modes of the wavetrains, one also needs to compute the so-called absolute spectrum of these wavetrains. This allows us to decide whether the instability is of convective or absolute nature. Only the latter case implies an instability of spirals in finite domains. We carry out this programme for the case of core breakup in an excitable reaction - diffusion system, the modified Barkley model. Our analysis yields that core breakup can result from the absolute variant of a novel finite wavenumber instability of the radial dynamics where the critical perturbations are transported towards the core. From these results, we can confirm that a simple spiral breaks up if the wavetrain in the far-field is absolutely unstable. The central new result is the discovery of respective convective and absolute instabilities of wavetrains connected with modes of finite wavenumber that propagate in the direction opposite to the wavetrain. Hence, in spirals, perturbations are moving inward and core breakup becomes possible even in the absence of meandering.