Important thing to remember is people's abilities vary as do what tools they have at their disposal. What is simple for me is next to impossible for the next guy and the opposite holds true in at least as many cases. That being said my standard procedure is predicated by A. Making all efforts to not twist fasteners off in the first place and B. not making the situation worse.
I have watched mechanics carelessly twist off a line of bolts or studs and then complain about all the work in from of them. Heat and beat and wiggle are the terms to remember. As one of the earlier posters said tapping on the end of a stud or bolt can do wonders for loosening it.
I might as well put my too cents in for penetrating oil. If you have weeks to fool around with the job at hand than spray on what ever brand you think smells the best as it don't make a tinkers damn what brand is on the can. All so-called penetrating oils are nothing more than various oils and solvents mixed at various ratios and some add perfumes to fool the masses. Blaster works best for killing wasps so I keep a can around, WD-40 keeps the wife happy so have at it dear. If I am working on somebody's car I use something that doesn't smell like it came from a sewer. The very best I have ever used, (which actually means it is no better or no worse than any spray bomb stuff) is better because it is virtually free. Dirty parts washing solvent. It is oil, plus solvent, same as spray bombs but no fancy can nor perfume. I have thought about adding perfume and canning with a highly colored label so as to get rid of it without paying a fee. Penetrating oil does have it's uses in my shop. Once you have a nut moving on a stud it rapidly gets in there and helps in removing it. :)Off my box about oils, please use whatever puts a smile on your face
So once you have sprayed it with oil, tapped it, heated it, sprayed it again, waited 14 days and no joy so you twist it off in a fit what to do next?
Welding a nut on works,,, sometimes. Other times it too twists off leaving you with a heat hardened stub that you can barely grind, and cannot drill with the bits found in 99% of tool boxes. Aka, a mess on your hands.
Easy outs, They have exactly three applications., A, studs or bolts that have been sheared off but are not really stuck in the threads. B. Heads rounded off and you cannot for whatever reason use a visegrip. C. Pipe fittings. Anything else they stay in the toolbox. Why? break one off and have that mess on you hands and come back later and report to us how you got it out.
OK so what does Butch the know it all do??? I drill to tap size and tap or pick the thread coils out of the hole, works EVERY time as long as it is done to start with, not in afterthought. There are two tricks to this that people either skip or don't know about and they get into trouble. A. is you must grind the end down to flush. I know, a hard thing to do when you have a stub still sticking out but get your die grinder or 4" grinder out and grind it flush, grind right down to where you lightly grind the casting so you can look, and see the center. The human eye has a remarkable ability to find center of a circle, ever shoot a rifle with a aperture sight? best open sight ever devised and works on same principle. B. Once flat take a sharp center punch and tap it leaving little more than a mark and stand back and look. If its not in the center you drift the mark by leaning the punch and tapping. Once you have it in the middle give it a good solid whack. Then get out your packet of new 1/8" drill bits and drill just enough to take out the punch mark, and stop, look to be sure your in the center before going onward. Then if your angle challenged have a helper stand back and tell you up/down/sideways/ what ever you must do to be drilling straight. It is easy to tell when you through the stud even when it is bottomed out as long as you have a sharp, good quality bit. This is no place for cheapskate and believe me I am as cheap as they come I buy 1/8" Champian black gold bits by the packet of 10 and use them exactly ONCE for this job. Once pilot drilled you go directly to tap drill size and if you have done your work very well you pick the threads out of the hole like a spring. If not you run a tap in a turn at a time and back it out removing trash from the hole as you go.