If I shot brick walls head on, I would. If I shot a lot of portraits where barrel distortion would create an unflattering look, I probably would. If I was using a lens with bad CA that showed up clearly in my photographs, I would.
For the most part, I don't find distortion or CA is ever a problem in my work, so I don't ever correct it. One way or another, lens correction diminishes detail, and maximizing detail is the most important thing for me. I'm happy to deal with invisible distortions and microscopic CA halos at the fringes of a landscape photo. Even though it may not be technically correct according to how the real world looked, you don't actually notice the difference until you toggle the correction on and off. In my bird and wildlife stuff, I've never noticed either to really be a problem except in a very few cases with CA.
As for vignetting, I actually like that as an artistic factor a lot of the time. I've never corrected it even once, but then again, I never really see it as I use APS-C. I guess if I picked up FF and experienced it more, I might correct it at times...depends on whether it has artistic appeal.