Abstract <p>We consider the problem of finding solutions to the Dirac equation for a free massive spinor field in the Schwarzschild spacetime. It is shown that the radial part of this equation can be reduced to the form of a confluent equation with four singular points and, apparently, its solutions cannot be expressed in terms of known special functions. In contrast, the radial part of the massless Dirac equation in the Schwarzschild metric can be reduced to the form of the confluent Heun equation and the corresponding solutions can be expressed in terms of confluent Heun functions. To study the asymptotic behavior of the solutions, they are found also in the form of infinite power series in the vicinity of the singular points of the equation using the Frobenius–Thome method. A doubling, compared to the case of Minkowski space, of the number of physically adequate solutions with energy greater than that of the particle mass has been observed.</p>