I think the problem is you are selecting only a single vertex. To separate you will need more than one vertex selected.
Under Edit Mode either manually select the whole spline, or only the desired segments before separating. Alternatively select only one single vertex from the Spline you wish to separate, and press Ctrl+L to select all linked vertex > Press P to separate whole segment.
You should now end up with another object that contains new spline.
Seeing as you already seem to have attempted separation by selecting a single vertex, you may need to do some cleanup.
Although apparently nothing happened, you probably succeeded in separating a single vertex curve object. These are very hard to select, and nearly impossible to see. In Object Mode either hide the larger curve objects, or use Border Select and Border Deselect to attempt to select the single-vertex curves and remove them.
On a side note, if by "styling" you simply mean assinging a different material , you can style each separate spline inside the same curve object individually by assigning it to a separate material slot, and choosing a different material for each slot.
You can also control bevel thickness on a per-vertex basis with the Radius property, Alt+S in Edit Mode for the currently selected vertex.