Is there a way in Capture to highlight selected nets?

If you want to highlight a particular net in the page, select that net, right click and select "select entire net", this will highlight that net on your design page.

If you want to track that net in your whole design, you can select the net, right click and select "Signals", this will open the signals window and you can navigate to that net by double clicking on the signals and it will take you to that corresponding page and highlight that particular net.
Is this what you are looking for ?

I am looking for a way to highlight selected nets in the whole design persistently. Signals is OK but it doesn't work for a group of randomly selected nets. It appears that what I want is not possible in Capture. How difficult it would be to write a custom add-on of some sort that would do it?

There is also the option to change the net properties which may give you what you need. Not quite highlight but a similar affect. Select the wire then right click - Edit net properties and choose a color, line style / thickness. Once you click OK this applies to the entire net across the design.

Unfortunately, this method is too slow. In my opinion, the "highlight net" has to be accessible directly from the menu that opens on the mouse right-click just like in e.g. old P-CAD 2006. It would also be nice to be able to open a list of all nets in the design and to be able to manipulate the net names and other properties from it, again as in the old P-CAD 2006, although I suspect there are even better and more powerful examples.

Unfortunately, this method is too slow. In my opinion, the "highlight net" has to be accessible directly from the menu that opens on the mouse right-click just like in e.g. old P-CAD 2006. It would also be nice to be able to open a list of all nets in the design and to be able to manipulate the net names and other properties from it, again as in the old P-CAD 2006, although I suspect there are even better and more powerful examples.

You can use Edit - Browse - Nets to adjust properties on nets in a worksheet but you won't be able to edit the netname itself, for that try Edit - Global Replace. I would also say that there is a post in the forum OrCAD Capture CIS productivity Improvement - Capture CIS - PCB Design - Cadence Community where you can ask for enhancements so make your suggestion there. 

Thank you for your suggestions. Global Replace is better than nothing but it doesn't even support wildcards and wouldn't let you go through what it has found before replacing. And why on earth is it separate from the main Find function?...

I was going to post something to that productivity improvement post in the past but hesitated as I felt that I would have to prepare a list of requests first. I should probably get it started.