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Has a ground and positive terminals. If you touch them directly together they make a beep. So you put one end on a set of leads, and put the other end on the other side of the leads to check if your traces / components are alright.
Just wanted to make sure he didn't go out and buy one of those automotive test lights you see at Canadian Tire or equiv.
There may be others around from Victoria also, I think Torben is from the island. Hard to say because a lot of people don't fill in their location.
Your router is most likely hosed. Without more information, such as a picture of visibly burned parts, model number, etc we will be unable to say if we can or cannot help you. The odds of fixing this remotely without a schematic are infinitesimally small.
I have seen this before with D-Link.
The reason for the Zener to explode is that it's just doing what it was put there for. Protect the router in case of too high input-voltage from the mains adapter. And this is where the REAL problem is. In many switched-mode adapters (the ones without an old fashion heavy transformer with an iron-core) electrolytic cap's go bad. That will happen in MANY, MANY of these adapters. The ASUS WL-500g router adapter is a good example and so are a lot of adapters for DVB-T receivers.
Take the adapter apart (if glued, put in a wise and apply some brute force and try to pry open with a flat screwdriver) and replace all big & small electrolytic caps. But first you check if the switching transistor (may be an Integrated Circuit) or any diode is shorted. In that case buying a new adapter is in order.
Now replace the dead zener inside the router (6,8V/1Watt in my opinion) and you're set to go.
If you are repeatedly unplugging, this would cause a cumulative effect, over time.
Get a Zener with leads on it, and solder to the solder pads that are in the oblong silkscreen between ZD1 and F1. Check for continuity across F1 before you go to radio shack you might need a fuse as well.