I have a small problem - how to select the points on the line but not select the points at the intersections of lines using "select by location"? Illustration below. I am using ArcGIS 10.0
This is a bit more complex, you will need to create two temporary 'unselect' feature classes. Use the tool Feature Vertices to Points with the 'BOTH_ENDS' specified to create points where you don't want selection. Do this for the blue and the red lines then merge.
Use Select by Location to select the points within a small tolerance from the blue lines (how far depends on your data) then using the selection option "Remove from the Currently Selected features in" select against the end points created earlier to remove them from the selection.
Select 1 (use a tolerance):
Select 2 (remove unwanted):