level strapping should run perpendicular to the joists and GWB should be installed perpendicular to the strapping.

It should run perpindicular to the joisting. Running the 4' side with the joists even at 16"o.c. doesn't allow enough support and can lead to buckling at the seams. Running perpindicular and staggering the sheets 4' is more "structural" and will give you a leveler lid. I've only seen strapping on the east coast and was impressed with it, but it's not necessary.

What is the purpose of the strapping? What advantages does the strapping provide vs just applying sheetrock directly to the joists?

In the West it is common to use GRN douglas fir which is dimensionally more consistent than the KD Hem-Fir or Spruce-Pine-fir used in the East, however, if you want the ceiling to be perfectly flat you would be wise to use strapping whatever the framing lumber might be.

In the Boston area almost all drywall is veneer plaster on blueboard (no taped joints) so the issue of where the square edge joints are located is irrelevant. You would think this would make strapping unnecessary but it doesn't.

If there is no ceiling trim, the edges of the ceiling at the walls should be held in place by the boards on the walls rather than being screwed to the ceiling structure in order to reduce cracking at this joint.

Strapping does not give you a flat ,level ceiling unless the framer/carpenters take the time to shim over/under crowned lumber. This is the case w/ dimensional lumber that is inconsistent strapping or not. Engineered lumber would be more consistent. I never seen shims used by framers back east, not to say that it isn't done and maybe is more so now days, just not back in the mid eighties when i was working back there.

Drywallers out here theses days are carrying power planes and shims to address framing members that are inconsistent before hanging the rock. 5/8" wallboard is more prevelant these days rather than just for fire code reasons.

I would give the plasterers over blueboard the credit for turning out a nice, smooth, level lid and walls, not the use of strapping.I would view one big advantage to strapping is extra structural to the joisting/ rafters to lessen the chance of twisting similar to the days when cross bridging was and maybe still is, used.

The sole purpose of using strapping is to be able to level the ceiling. To use it and not level it would be pointless and a waste of material and labor.

Veneer plaster is simply too thin to disguise unlevel GWB. It is not applied like a full plaster coat, it is a veneer. And it is provided by the drywall trade rather than by plasterers. Using plasterers would double the cost of installation.

You must think contractors in the East are looking for ways to waste money.

I dont feel new england builders are wasting money, they are following tradition. A lot of the techniques used have been in place for centuries. I respect that and was impressed that you generally only see that in and around the new engalnd states. It doesn't make it right or wrong,better or worse, it's simply traditonal. If it aint broke, dont fix it. I always will consider the tradional exterior/interior trims and design on new england "pisser"! The majority of homes and design are traditional coinciding w/ todays materials. I simply stated that fastening strapping tight to the underside of dimensional joists/ rafters isn't going to give you any flatter/leveler lid than not using it. It is up to the eye of the framer/carpenter to pick out decent,straight lumber to be used in those areas and setting aside warped,twisted lumber generally cut up and used for blocking. The only way to achieve a true lid would be to plane or shim coinciding w/ the strapping install as well as the use of laser levels or string lines. Nobody does that as a rule because home builders are building homes, not pianos. Again a decent framing job is all about the effort made by the tradesperson. When we were roofing/siding homes on the cape and around boston, strapping was explained to me that as the top of the joist/rafter is kept in place by the subfloor/sheathing, the bottom side is kept in place by the strapping. No mention of it's sole purpose being to level out a lid. Common sense dictates that the strapping will follow whatever the framing member is doing straight , bellied, bowed, or otherwise. That's why picking out straight material is crucial for a clean project same as layout and cuts are critical to try and accomplish a dead on frame. Engineered lumber has made the process easier, but even w/ shimming/planing, wood can still move so in that respect, there is no perfect frame project or true,dead on walls and ceilings.

Now that there has been all this speculation. The reason for strapping a ceiling is.
1 It ties all the floor joist together so you don't have them moving independently from each other when some one walks on the floor above, thus cracking the sheet rock.
2 It adds bracing to the floor joists.
3 It takes out some of the unevenness of the floor joists.
4 It gives a wider surface to hang the rock on and a better surface to screw at the joints. Sheetrock does not like to be fastened less than 3/4 of an inch from the end. Strapping gives you an 1 3/8 inches to screw at the joints.
So to answer the question the sheet rock should run opposite the framing the board is not strong enough to only be screwed twice in the field on a ceiling (or 16inches on center)

Thanks for the info,d.p.

Bracing, bridging or blocking adds no important twisting or deflection benefit until the joists are larger than 2x12, but it was thought to do so in past centuries and is still done in some areas of the country even thought is has been removed from the IRC for 2x12's or smaller joists. Even when required, it only needs to be spaced 8 ft. on center.