For a single hadron bunch in a circular accelerator at zero chromaticity, without multi-turn wakes and without electron clouds and other beams, only one transverse collective instability is possible, the mode-coupling instability, or TMCI. For sufficiently strong space charge (SC), the instability threshold of the wake-driven coherent tune shift normally increases linearly with the SC tune shift, as independently concluded by several authors using different methods. This stability condition has, however, a very strange feature: at strong SC, it is totally insensitive to the number of particles. Thus, were it correct, such a beam with sufficiently strong SC, being stable at some intensity, would remain stable at higher intensity, regardless of how much higher! This paper suggests a resolution of this conundrum: while SC suppresses TMCI, it introduces head-to-tail convective amplifications, which could make the beam even less stable than without SC, even if all the coherent tunes are real, i.e. all the modes are stable in the conventional {it absolute} meaning of the word. This is done using an effective new method of analysis of the beams transverse spectrum for arbitrary space charge and wake fields. Two new types of beam instabilities are introduced: the {it saturating convective instability}, SCI, and the {it absolute-convective instability}, ACI.
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