The mode spectrum associated with a magnetic X-point is determined, taking into account resistivity and electron inertia. The assumed equilibrium configuration has zero current. The eigenmodes are expressed in terms of solutions of the hypergeometric equation, and are also computed numerically. The spectrum is shown to exhibit a rich structure of continuum and discrete modes. When both resistivity and electron inertia are present, there are two distinct continua associated with singular current sheets and infinite kinetic energy, corresponding to zero and finite real mode frequencies. A finite number of discrete modes can also exist, depending on the Lundquist number and the collisionless skin depth.