For arbitrary closed-line magnetohydrostatic equilibria the continuous spectra are studied which correspond to singularities not in the direction of the pressure gradient. Unlike the familiar Alfvén and cusp continua, these new continua may be unstable. Their stability is governed by the closed-line version of the ballooning-mode criterion, and the corresponding growth rates are the eigenvalues of an ordinary differential operator of fourth order (rather than of second order).