Removing vertices causing self intersection or examining a multi-part polyline can be easier if the modify features panel is used. 

Start an edit session and select the polyline in question using the select tool.

Click on the Edit vertices button (Edit Vertices tool) in the Tools group on the Edit ribbon to display the individual vertices of the polyline. Vertices will display as green squares.  The red vertex is the To-Node.

In the image below, selecting a polyline and choosing to modify the feature through editing its vertices turns on the visibility of its vertices. In the modify features panel selecting a row (e.g. row 5) highlights the vertex as a solid green square.  Using the up and down key on your keyboard you can step over vertices

Modify Features Panel

Right click on a vertex in the modify features panel will display a pop up menu offering a variety of actions, including zoom to, flash and delete vertex.

Holding the shift key down when selecting a vertex via the modify features panel will allow you to select multiple vertices, all of which are displayed as green boxes. 

If your data was Z or M enabled you would see these columns in the modify features panel.  If your data is multi-part then the vertices listed will be grouped into numbered parts.

As you have seen, the modify features panel is a very useful way of looking at your data on a vertex by vertex basis.