The nature of local thermal instability in static and dynamic radiating plasmas described by an equilibrium cooling function has been reexamined. Several new results have been found. In a plasma in both thermal and hydrostatic equilibrium, if the cooling function is not an explicit function of position, and does not display isentropic thermal instability (i.e. sound waves are thermally stable), then isobaric thermal instability by the Field criterion is present if and only if convective instability is present by the Schwarzschild criterion. In this case, thermal overstability does not occur. For the case of a dynamical plasma we present a very general Lagrangian equation for the development of nonradial thermal instability. In the limit of large cooling time to free-fall time ratio, the equation is solved analytically by WKBJ techniques. Results are directly applicable to cluster X-ray cooling flows. Such flows are surprisingly stable except for perturbation wavenumbers that are very nearly radial. We believe that the origin of cooling flow optical filaments is not to be found in linear thermal instability.