How durable is a supercapacitor?
Overview

In general capacitor lifetime (including supercapacitors) is dependent on three things: 


Electrolyte Life
Voltage Derating
Temperature / Power Dissipation


If you want the capacitor to last a long time, limit the applied voltage, keep it cool, and limit the output current.  All this should be in the datasheet of your capacitor.

Electrolyte Life

If the capacitor is, say, ceramic or tantalum, the electrolyte is a solid and the cap will basically never go bad.  If it's an electrolytic, then it contains fluid which will evaporate and eventually cause the cap to fail.  In an electrolytic double-layer super-capacitor, the electrolyte is a combination of a fluid and of activated carbon, so it is mildly vulnerable to evaporation.

Voltage Derating

More important, though, is the voltage derating of the capacitor.  If the voltage burns the cap up on the first use, you won't need to worry about evaporation.  A capacitor is a carefully constructed device which separates two conductive films with a thin, thin layer of insulating material over a wide area (folded or rolled into a package). Decrease the separation, and you've got higher capacitance with the same area.  This thin separation is vulnerable to high voltages; that's why capacitors have specific voltage maximums, often printed on the case.  In a supercap, this barrier is often just nanometers thick, and the dielectric will not insulate high voltages across this short distance.  

Using a cap at close to its maximum voltage will cause it to fail more quickly than using it at a lower voltage, this tradeoff is known as a derating curve.  It should be available from your capacitor manufacturer.

Temperature / Power Dissipation

A capacitor is negatively affected by heat.  It causes the electrolyte to evaporate more quickly, causes the dielectric to be weakened, and it can damage the thin conducting elements in the capacitor.  Both environmental heat and self-heating effects should be considered.  If the capacitor is discharged very rapidly, the small resistance of the foil and leads will be inconsequential compared to the square of the current.