In most photos, according to the subject and genre of photography which you choose, having sharper edges/details is desired, but I am not sure if there is a different process for different genres or not.
The short answer is: yes, sort of.
I don't think there's a absolute rule for sharpening in any given genre, but there are some rules of thumb that may get followed from time to time. For example:
Portrait photography typically likes to focus in on the eyes and soften out skin blemishes such as scars, acne, etc. So, your sharpening technique there is probably going to want to reflect some of that. Also, portraits are often seen as more pleasing when they're a little softer (not blurry though). Except the eyes. Sharp eyes are good. :)
Nature photography has a fair amount of variance in it. Animal shots usually want to be sharp, as with general landscape. Water, though, people like seeing it blurred through long exposure, same with fog. So, in these case, the amount of sharpening depends on the main subject and, as with portrait, you might be more targeted.
Architectural and product photography usually want to show the detail and so overall sharpening is probably desirable.
However, I wouldn't probably spend a lot of time worrying about any of that too much. Like all forms of post processing, the best approach is to maximize your enjoyment of the image. Some times that may mean a lot of sharpening work, sometimes very little. I wouldn't let these ideas get in the way of making your photograph.
The question I linked to in the comments really does give you some good information that's going to apply in general. Couple that with the masking features of tools like Photoshop in order to apply your sharpening efforts to controlled areas is really what you need once you know what you want to sharpen and why.