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Yet another example.  If using MOSFETs only for power switching applications, it can seem like these are "robust" and "immune" to ESD events.  While the package clearly states "ESD sensitive", devices never seem to fail from ESD - why?

Because power MOSFETs have a very large gate and other capacitances - sometimes as high as hundreds of nano-Farads.  So an ESD event is complete (a few microseconds) long before one of these pins has a chance to exceed the max rating of the device. The whole device just charges slightly, thanks to the capacitance.

The venerable 2N7000 series of MOSFET however (low power, fast switching) have very tiny capacitances, so are thus very "sensitive" to ESD events.  Simply because a "zap" to any pin allows the voltage to rise quickly without capacitively coupling to the other pins.  Out of all common jellybean parts, these might be the *most* sensitive, and can certainly fail just from being handled without proper ESD protection.  As the other answers indicate, "failure" can be anywhere from "total short" to "slight degradation of performance." In general, the faster the MOSFET, the lower the capacitances, so thus the more the ESD susceptibility.  Same for IGBTs and any other silicon - lower capacitance = higher susceptibility.

I have a chair which generates static electricity (triboelectric effect) whenever I get up from it (in dry air.) I can get up, walk over to an electronics workbench with ESD mat, and "zap" a board sitting on it (due to the capacitance of the board.)  This one zap could certainly destroy something. That *could happen* if I were *careless*.

I could instead walk over to it, touch both hands to the mat, then touch the board with no issue whatsoever.  The charge would be quickly drained from my body due to the slight conductivity of my skin and the mat.  So part of "ESD Protection" is being cognizant of what ESD is, how / when / where it is generated, and how to ensure you won't accidentally zap something.  Even consider that your backside clothing could still have a charge on it (especially true if wearing something like a jacket), and backing up to the bench could touch this to the board and cause damage - even after discharging yourself.  

So yes, it is best under any condition to sit down at an ESD-safe workstation, wear the high-resistance-grounded wrist/ankle strap, and then perform the ESD-sensitive work.  This gives relatively good protection from most conceivable ESD events. Barring that, just being *aware* and *careful* goes a long way also.  If assembling one-off prototypes or the occasional hobbyist board, a full-fledged ESD workstation is impractical.  Likewise, large-volume PCB assembly houses all have and use ESD workstations, as +2% failure rate due to ESD (really, operator laziness/ignorance) would cut into their already tiny profit margin.