Mainly varies with rating, and somewhat with technology. In my experience, newer SuperJunction types are fully saturated by 8V or so -- they don't draw additional current, even under fault conditions (Vds > 100V, well above the resistive region), at higher Vgs(on). This is I think unusual, and probably unique to the technology.
Most classic MOSFETs keep increasing with Vgs, at least by 15V, not that you'd have any use for that outside of pulsed applications as the voltage drop is so high. Likewise in most IGBTs, high Vge(on) is problematic under fault conditions, as the die is so small, only a few microseconds of fault are permissible before destruction occurs.
Most industrial modules (especially IGBTs, maybe not so much MOSFETs?) specify not only 15V on, but -15V off. I think that's the most extreme case.
These are all high voltage examples (Vdss > 300V). In the other direction, low voltage devices offer higher transconductance and logic-level inputs. I don't think there are many (any?) "logic level" types over 250V or so, but, below that they are reasonable. Typically a logic-level type is distinguished by lower Vgs(th) of course, as well as higher Qg -- no free lunch, that means you need maybe a fourfold stronger driver to get the same switching speed. In still lower voltages (under 100V say), most devices are rated for 5 and 10V operation anyway, and low voltage devices (30V or less) may even be rated for 3.3 or 2.5V. Very low voltages (<= 10V) are even available for 1.8V, or 1.2V I think. Such low voltage devices of course carry a Vgs(max) of 12, 8, even 6V or less.
There is some wear effect of high Vgs(on) and low Vgs(off), near ratings, but I think it's normally specified to a reasonable lifetime (maybe a decade at max temp and rated voltage?), so it's not fatal to run near it, but it's preferred to run lower.
The failure mode beyond that, I believe is more to do with flux (voltage * time) spent at the extreme, as tunneling current is drawn through the gate oxide (some ~uA even at room temp), which can trap charges in the oxide, shifting Vgs(th) and eventually causing breakdown. Sudden breakdown can occur anywhere from 30V to over 80V, for a typical 20V rated gate.
Tim