If you are trying to reduce the number of vertices in a feature, you might want to check the simplify tool in the cartography toolbox. It can remove extraneous vertices while keeping the essential shape fo the feature(may or may not be what you are needing.
ArcGIS Desktop
Another possible tool is the Generalize tool in the Edit Toolbox.
ArcGIS Desktop
If you are not trying to remove the points, just identify them, you could convert vertices to points then do a near table to see which points are spatially closest to each other.