"Broken" nuts are generally pretty rare, at least in auto or aircraft maintenance, but sloppy use of ill-fitting wrenches (spanners for UK people, I guess) can easily take the points off, making it nearly impossible to turn the nut. It's also fairly common to find nuts rusted onto the bolt/stud so tight that you may break the bolt if you put enough torque on to do anything.

The well-equipped mechanic generally keeps a "nut-cracker" that can be slipped over the nut, and when you turn the screw on the side of the cracker it drives a wedge into the side of the nut until it separates enough to be loose on the bolt, or in really tough cases, cut the nut on two sides so a goodly piece falls out. Lug nuts are usually wider than others, so an oversize nut-cracker might be a good investment for anyone working in a wheel shop.

The standard size is generally all you'd need for working suspension parts, where frozen nuts are probably most likely (since they're exposed to all the road slush there).

Most shade-tree mechanics can get by with "knowing someone who has one," since there really are few occasions when you really need one, although they're not particularly expensive.

As to the right-handed vs left-handed threads, I think there's a rude song or two, and a couple of limericks, on the subject, but since we're below the line I won't bother to try to quote one here.

John