Why can I not select any vertices at all? I have been trying to select any vertices, faces or edges. It refuses to select anything.

I have selected the required selection on the bottom of the screen for vertex selection still it refuses to let me do it.
All my settings are defaults. I am stuck, no matter what I do!

I clicked LMB, RMB, even clicked down on my scroll, No success.

Why can I not select any vertices?
This is weird. Check the following:


The selected Object is a Mesh (not a Lamp, Camera, Empty or other form of weirdom).
You are in Edit Mode. To check this, look at the lower part of the 3D-View for selectable box with the text Edit Mode or Object Mode. If you hit Tab (with your cursor in the 3D-View proper), it should switch between Edit- and Object Mode.
Hit A to select / deselect all vertices. 
IF no visible change happens, you are doing art. Else, it is the vertexial selection that fails. 


Hope this helps.