What mode is your camera most often in? Why?

In answer to the explicit question in your OP, I use aperture priority most of the time, manual when I need control of the shutter as well, shutter priority once in a blue moon, P never. But let me try also to answer the implicit question in your OP as to why many use aperture priority.

With a given lens and a given framing of a shot, there are only two things you need to control with your camera: exposure and DOF. [I'm assuming you're shooting raw, so WB is not a consideration, and I'm ignoring the subtleties of the effects of ISO on noise, which would take all this too far afield.]

The exposure is determined by the aperture (here meaning F-stop) and the shutter speed. The DOF is determined by the aperture. Aperture, then, is the one camera setting that affects both DOF and exposure. So using aperture priority allows you to control the DOF while allowing the camera's metering automatically to select the shutter speed needed to get a (hopefully) correct exposure. You thus get all the control you can have over your camera combined with programmed use of the camera's metering.

The above works well much of the time, and that is why aperture priority tends to be used by many.

But, it doesn't work all the time. If the light level is too low, the metering may select a shutter speed that is too slow, thereby allowing camera shake and/or subject movement to become problematic. Both can be helped by increasing ISO to allow the camera to pick a faster shutter speed. Or hand shake can be helped by bracing the camera or using a tripod.

The above may also not work well if an inappropriate spot is read by the metering. Metering the shadows can lead to overexposure; metering a bright spot can lead to under exposure. There are several ways to get around this (that are useful in S priority as well). One is to use AE lock to let the meter read and lock on an appropriate part of the scene before recomposing to take the shot. You judge the appropriateness of the exposure using the histograms. Another is to use EC to compensate for an inappropriate meter reading, going to positive values if you are getting underexposure and negative values with overexposure.

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gollywop