In transonically rotating toroidal plasmas a new class of local magnetohydrodynamic instabilities is found, called trans-slow Alfven continuum modes, that are due to poloidal flows exceeding the critical slow magnetosonic speed. When this condition is satisfied, virtually the whole plasma becomes unstable with modes localized at, or close to, rational surfaces with approximately the same growth rates. The instabilities are studied from a general point of view, treating magnetically dominated plasmas (tokamaks) and gravitationally dominated plasmas (accretion disks) on an equal footing. In the first kind of plasmas, rotating overstable modes are found with growth rates that are a fraction of the Alfven frequency, determined by the poloidal Alfven Mach number. When the mass of the central object is increased, these modes lock to become explosively unstable with growth rates that may exceed the Alfven frequency. The instabilities are localized on the magnetic/flow surfaces to which the flows and magnetic fields are confined as long as no anomalously large dissipation mechanism is present. It is suggested that the trans-slow Alfven continuum modes may generate the necessary turbulence to break this confinement so that accretion could take place and jets could emerge. This theory features: (1) a new effective toroidal scaling of the transonic equilibrium equations, (2) a new compact formulation of the equations for the transonic continuous spectrum, and (3) a completely explicit analytical as well as numerical investigation of the poloidal mode couplings involved in the instabilities. (C) 2004 American Institute of Physics.