It is well known that singular configurations are inherent to parallel manipulators and have serious influences on their properties. Therefore, these singular configurations should be avoided in the design and application of mechanisms. The researches on the singularity identification and distribution have revealed the relations among the six configuration parameters at singular points. Few works have dealt with the relation between the singularity and the input parameters, as well as the properties of the manipulator nearby the singularity. In this paper, taking the semi-regular hexagons 6-6 Gough-Stewart manipulator (SRHGSMP) as an example, the configuration bifurcation characteristics going with the input parameters, the assembly configurations at singular points, and the reasons to cause the singularity are analyzed. The research reveals that the number and the combination of the input parameters have great influences on the complexity of the singularity and the curvature radiuses of the configuration curves. Under different number of input parameters, the dimensional-utmost singularity, line vectors correlation singularity and Jacobian matrix correlation singularity can occur individually or jointly. Choosing the adjacent input parameters, the simple singularity and the large singularity-free input parameters zones can be obtained. And selecting multiple input parameters, the self-motion regions and the singularity avoidance errors can be reduced. These new discoveries are valuable and of significance for the trajectory design, the singularity avoidance, and the self-motion control of the parallel manipulator.