We present details of calculations of the core structure of a screw dislocation in a smectic A liquid crystal, using the phenomenological Landau-de Gennes free energy functional. The order parameter frustration created by topological constraints far from the dislocation core is resolved in one of three qualitatively different ways. The three types of dislocation core solution are the DT (double twist), CL (classical), and BP (broken polar symmetry) solutions, respectively. The stability requirements for these structures are discussed, as a function of temperature, smectic elastic properties, and coupling between smectic and nematic order. The effect of possible inhomogeneity between left- and right-handed conformers is also examined.