Shutter priority is not one that I use often. There is nothing wrong with it I just never really used it. After taking this series of photos I will make an effort to use it. I have seen the times that it could be useful. Such as when being more worried about stopping motion in the image as opposed to having a certain amount of the photo in focus. You control the shutter speed and ISO then the camera sets the aperture (or tries to within the lens limits) to what would make a "perfect" aperture using the exposure triangle.
If you are taking pictures of cars in motion, people running or playing sports, you have an artistic idea that has some motion blur and you want to control how much is blurred, you are trying to make a busy place look empty, taking pictures of fans, etc, then this would probably be the way to go. It can be used for all of those and more.
These images were taken at 800 iso with a tripod mounted camera. The first was and last images had a blinking aperture which means each was outside of the "perfect" exposure. 1/640 sec was a stop below what the aperture could adjust for and 1/2 second was a stop above. If you notice the first one is dark compared to the others and the last is brighter. Again we can see the triangle at work in these captures. For each stop (keep in mind these are full stops) the the shutter speed changes the aperture changes one stop too. I know that Sony and Canon have the option, not sure about Nikon I haven't used them enough, to change the stop adjustment to either 1/3 stop or 1/2 stop when changing the shutter speed or aperture. I hope this collage provides a good example of shutter speed adjustments. Now go out and shoot.