How do I create a single vertex (not connected to another vertex) on the cursor position?
The "extra objects" addon (included in the latest release of blender) has a add object> single vertex option. See the second half of this answer to learn how to enable the addon. the Shift+A>Single Vertex.
To expand on @cegaton's answer:
After creating a vertex with Ctrl+Left Mouse, you can use Shift+S > "Selection to Cursor" to move the vertex to the 3D cursor's position.
If you want to move several vertices to the cursor position but keep their relative shape, use Shift+S > "Selection to Cursor (Offset)"