Although the space charge typically suppresses the transverse mode-coupling instability (TMCI), it does not mean that it stabilizes the beam. Suppressing TMCI, the space charge brings in a whole family of convective instabilities for the same beam and machine parameters. This new sort of instabilities will be introduced and generally described. They explain tail-dominating oscillations and weak dependence of the transverse stability threshold on space charge, observed at CERN SPS.