My personal experience is similar to Kevin54's with a few exceptions.
I can relate to the "get the tap out after someone else has made it worse" senerio. That said......
The punch method of shattering a tap can work sometimes. Depends on the work piece, the tap, ect. Not good for everything but has it's place.......kinda hard on the punch too.
Like most things, there are many ways to do something, the trick is finding a way that gives the desired results without adding to the damage (think hypocratic oath). If the person with the broke tap is unsure of their skills, it's best to let a pro look at it.
Welding is usually a good option, and cheap.
I've used the torch method of melting a tap, but only on through holes. Blind holes don't sound too good for this method.
EDM's are cool and all, but like Kevin54 said, finding one, and then finding someone that can use it is the problem.
Carbide endmill in a Bridgeport is a good method if you have the means.
Those Walton tap extractors are a pain and usually add to the frustration. I've never had good luck with them, and certainly never enough luck to warrant buying a set for myself. Some will dissagree and that's fine. They probably work ok for hand taps broken
by hand and not taps broken by machine (power driven).
Just my .02.
