What are the best roof ventilation methods for different roof shapes?
Long gable ridge with deep overhangs
The textbook case. Vented soffits along both eaves feed a ridge vent along the peak, giving even airflow under the whole deck. Close off gable louvers so they do not short circuit the ridge.
Hip roof
Hip roofs have intake all the way around but short ridges. If the ridge is too short to supply the needed exhaust, the options are box vents placed high on the slopes, specialty hip vents where approved, or a combination calculated so exhaust does not exceed intake. Mixing a ridge vent with box vents is usually discouraged.
Little or no overhang
Some newer and many older homes have almost no soffit. An intake vent installed at the roof edge, under the first courses of shingles above the drip edge, can supply air where there is no soffit to vent.
Vaulted or cathedral ceilings
Each rafter bay needs a clear channel from eave to ridge, usually with baffles and at least 1 inch of space above the insulation. Where that is not possible, an unvented assembly with insulation applied directly to the underside of the deck, built to the code's unvented attic rules, is the alternative.
Complex roofs with many sections
Dormers, bump outs and separate attic pockets each need their own intake and exhaust. A ridge vent on the main roof does nothing for a sealed attic over the garage.
Where powered fans fit
Rarely as a first choice. They need generous intake and can pull conditioned air from the house. See ridge vent vs box vents and roof ventilation basics.