I have an image made up of several shapes. Some will show nodes, others will not. Shapes that won't show nodes can be edited by pulling arrows, but need to change in more detailed ways.

How do I get the nodes to show for editing?

If objects are on a locked layer -open the layers tab and unlock them
-but as you can move them, they are not on a locked layer.
They are not even locked objects that could be unlocked under the object menu's unlock all objects option.
By selecting them, the bottom line of the window-indicator line tells what is the selected object.
As you cannot further ungroup those, they are not a clipped or masked objects.
Maybe they are clones of a parent object -release clone link by pressing Shift+Alt+D.
With this explanation it's unlikely that they are part of a pattern fill that I previously thought.
They might be raster images.

If in doubt, open the xml editor and see what are those objects.