The operational stability of a composite tape/film-type superconductor is studied by considering the impact of a thermal disturbance distributed along a line. Issues addressed are failure of intrinsic stability, current sharing, Joule heating, and recovery of superconductivity. Mathematical models have been developed to describe the phenomena. The conjugate heat diffusion problem for the superconductor and the stabilizer has been solved by the finite-difference method. The effects of current density and stabilizer dimension have been investigated, resulting in the identification of three modes with regard to the operational stability of the superconductor: intrinsically stable, recoverably unstable, and irrecoverably unstable. Numerical data of instability parameter, convective cooling rate, Joule heating rate, and current sharing are presented for NbTi superconductors.