A two-dimensional criterion for distinguishing absolute instabilities with arbitrary propagation in an infinite magnetized plasma is developed and applied to mirror-confined plasmas. It is shown that this criterion follows directly from the one-dimensional criterion by careful consideration of the singularities in the inverse Fourier-Laplace transform integrals.A number of instabilities, generally referred to as loss-cone instabilities, are numerically investigated using the above criterion and the results compared to those available from one-dimensional calculations where growth is restricted to the direction parallel to the magnetic field. It is found that for the loss-cone instability the one-dimensional calculation gives good estimates of the transition density and real parallel wave number as a function of the electron temperature, but yields an overestimation of the maximum axial convective growth rate. When the effects of ion temperature anisotropy are examined it is shown that the two-dimensional calculation is required because of the great and sensitive dependence of the transition density on the value of the perpendicular wave number.Although the stabilizing effect of increased electron temperature on the loss cone, drift cone, and drift cyclotron instabilities is confirmed, it is found that this effect is smaller in the case of the drift-cyclotron mode. Moreover, the presence of a cool ion component in a hot loss-cone plasma is shown to have only a small stabilizing effect at the first ion cyclotron harmonic.The Dory-Guest-Harris instability associated with the ring distribution is shown to be absolute for flute-like modes, while for oblique propagation it is shown that an absolute instability cannot exist.For all modes, the one-dimensional calculation gives somewhat pessimistic results regarding the dimensions of a stable plasma. By allowing for spatial decay normal to the magnetic field the two-dimensional investigation shows that the plasma need not be as "stubby" to be stable against the loss-cone modes.