LANDSCAPEThe most obvious genre of photography that uses depth of field for maximum impact is landscape photography. This is one of the cases where stopping down and using the hyperfocal distance is important. Shooting a landscape wide open is making unnecessary concessions with your image sharpness. There's no real reason to photograph a standard landscape at less than ƒ/8 unless you have no tripod or you're isolating a feature of the scene.
ARCHITECTUREAs with landscape photography, when executing architecture photos, using a smaller aperture to create sharp images is preferable. The key is to show characteristic detail of the structure. Unless you're going for artistic detail shots of particular facets of the building, it's best to keep the aperture small.
ENVIRONMENTAL PORTRAITSThese are portraits of a person, which are generally used in documentary assignments where it benefits the viewer to see the subject in his or her home or work environment. While a typical three-quarter-length portrait of some-one may be great with an indistinct blur in the background, if the goal of the image is to tell a story about the person appearing in the photo, allow enough of the surroundings to be sharp enough to be recognizable.
STREET PHOTOGRAPHYThis type of photography is all about the environment and the subject's role within it, so getting things in focus is important. This brings up a challenge: how to get great street images with deep focus while still maintaining the subject as a key focal point and avoiding an image that appears chaotic. The solution requires a keen eye for composition, composing the frame perfectly so the subject and the important parts of the scene are in frame and in focus while making sure that extraneous details are left out.
Street photography is undergoing a massive resurgence right now, and I notice that the "shoot wide-open" technique is coming along for the ride. While I agree that it does sometimes have artistic merits, and in many cases works great for getting rid of distracting elements, it easily can cause potentially important and interesting details to fall by the wayside. In my opinion, shooting wide open and stopped down both have their place in street photography, but shooting wide open should be the exception rather than the rule.