Because of its reputed benign nature, the saddle point of attachment has not received critical attention to nearly the same extent as has the saddle point of separation. Recently, Visbal calculated low-speed flows around a cylinder mounted on a flat plate. Here, it was fully to be expected that the outermost critical point in the surface flow pattern ahead of the obstacle would be a saddle point of separation. The results indicated that the critical point was actually a saddle point of attachment, not separation. These results have brought to light a number of issues requiring additional study. In the present study, two numerical codes are used for a wide range of Mach numbers, Reynolds numbers, grid sizes, and numbers of grid points to confirm the existence of the saddle point of attachment in the flow before an obstacle. The computational results near the critical point are theoretically analyzed. The impact and significance of the saddle point of attachment to the interpretation of experimental surface flow patterns and the definitions of lines of separation and attachment are discussed. A line of oil accumulating from both sides can be either a line of separation or a line of attachment, depending on the characteristics of the saddle point.