I can't select individual points on a shape layer in Photoshop CC

I have the same issue, in 2020. Sometimes a shape layer simply does not allow one to select the anchor points - it's the craziest thing - it simply does not repsond to being clicked with the direct selection tool, white or black. What I've found works - facepalm - is to close the document, open it again, click on the layer and then suddenly the path and anchor points are available again.

I had the same problem in Illustrator CC: went to View/Show Edges. that displays the individual points.

I think I figured it out through trial & error:  to select only 1 node of a line (a line being 'node-line-node'), you have to dig deep into the hierarchy, and select 'Path 1' as an example - that allows easy node moving - now I can make lines any length and join them visually.

When you edit a live shape, it must be converted to a normal shape object. The only way to select points is to use the direct selection tool.

It's a bit of a black art when you start:

1. Lets say you have created a live shape layer, or a work path — the process is the same either way.
2. Hit the 'a' key to select the Path selection tool (black arrow) or Direct selection tool (white arrow).
3. If you have the black arrow, and click on your shape/path and move the cursor, the entire thing moves and you start to get annoyed.  Everyone has done this ;-)
4. Hold down the Ctrl (Cmd) key, and the black arrow will change to the more useful white arrow. Click outside the shape/path to unselect everything.  Now drag a widow over a small area, and just that point will be selected. You can now drag that point.

To give yourself more control, select the Pen tool.  You can switch to the white arrow at any time using the Ctrl (Cmd) key.  If you select a single point, and want to change it from a curve to corner point, hold down the Alt (Opt) key and click it.  If going from corner to curve, hold down the Alt (opt) key, and click and drag.