A pentapod is usually defined as a 5-degree-offreedom fully-parallel manipulator with an axial spindle as moving platform. This kind of manipulators have revealed as an interesting alternative to serial robots handling axisymmetric tools. Their particular geometry permits that, in one tool axis, inclination angles of up to 90 degrees are possible thus overcoming the orientation limits of the classical Stewart platform. This paper presents a solution to the problem of finding those changes in the location of the leg attachments of a pentapod that leave its singularity locus invariant. Although the solution to this problem does not provide a fully characterization of the singularities, it provides a lot of insight into its nature. It is shown, for example, that there are four different architectures for a pentapod with a completely different behavior from the point of view of their singularities. The kinematics of pentaponds with coplanar attachments at the fixed base has previously been studied as rigid subassemblies of a Stewart platforms. In this paper, we treat the general case in which the base attachments are arbitrarily located in 3D space.