In general terms, as you ask, well... there is no general answer to this at all. It entirely depends on the lens design, the sensor size, and, possibly, on obstructions mounted to the front of the lens like lens hoods and filters.
For example, my phone has a 3.8mm focal length, and shows no sign of darkening in the corners. However, the Canon 1200mm lens — that is, 1.2 meters! — has noticeable vignetting when used wide open. Or, herehere, where you see black corners with an image taken with my 23mm lens, simply because I incorrectly used the lens hood for my 56mm.
On the other hand, if I put either of my lenses in front of the sensor the size of that in my phone camera, the corners would be fine — and that's true of the 1200mm as well. But if I put my phone camera's lens in front of a DSLR full-frame or APS-C sensor, you'd just see a tiny circle in the middle and mostly "black ring".