It's rare when I'm NOT using Auto-ISO. My reasoning is that all three legs of the exposure have a dual-nature: all three have a "Virtue (something that works FOR us) and a "Vice" (something that works AGAINST us). These six attributes are the "balls" of the juggling act that is photography. No matter how far you progress with your photography you will never not be performing this juggling act. In time, you may have to think about it less, but the juggling act will never cease to be happening since it lies at the very core of photography itself. Shutter speed's vice is undesired motion blur. Aperture's vice is undesired softness or a flat out blurry shot and ISO's vice is digital noise.
Because I like to creatively control motion blur (or the lack thereof) and since it's almost impossible to correct unintentional motion blur in post processing, I like to be in charge of the chosen shutter speed. Because I like to creatively control the background, prefer sharp shots generally speaking and because it's almost impossible to correct a soft or out of focus shot in post processing, I like to take charge of the chosen aperture. Digital noise, on the other hand, is very correctable during post-processing and, at least in my opinion, its most potent vice does the least "damage" to a shot, comparatively speaking, whereas the other two typically result in an unrecoverable shot.
Yes, there are times when ISO is a much larger factor in the equation and when that is the case, I take charge of ISO as well. But again, generally speaking, ISO is the thing I worry about least which is why I'm comfortable letting the camera handle it for the most part. I can also constrain how much latitude the function has in the Auto-ISO menu as well so, in short... I don't really see any downside to using Auto-ISO.