You need to be an expert to predict when you don't need decoupling caps...
I vote this answer as the forum winner for the day!
Yep. We can explain why -- but honestly, it's not an explanation the beginner will get much use out of, so it's just as well to leave it well-enough-alone, as above...

It's not something you need to take on faith, but it's also not something that can be explained instantly. The gap between must be bridged with knowledge.
(So of course, I'll go on to explain, fully expecting the zooming "over my head" sounds from readers. I know, sorry.)
In short, the problem is not just transient (see above examples, like use as a comparator), but dynamic. It is a problem of stability.
The amplifier responds to differences in voltages, not just between its inputs proper, but between its power pins as well (this is measured as PSRR, Power Supply Rejection Ratio).
We can make the amplifier proper, into an oscillator, by connecting a feedback circuit between its input and output.
Well, if the power pins act as inputs as well, we can also construct an oscillator by connecting a feedback circuit in its power pins. The power pins are both input and output, so it's not obvious if we can get [un]lucky enough for this to oscillate -- but, just whether it's possible, absolutely.
In fact, transient reasons aren't very interesting to general purpose op-amps: the output might change state in microseconds, and might deliver tens of mA in that time. That's a slew rate of 10mA/us. Current rate times inductance equals voltage drop. The inductance needed for a significant voltage drop (say 1V) is a whopping L = V / (dI/dt) = 100uH. That's a
long power cable (about as many meters, give or take)! And that would just be a momentary dip, it might not necessarily oscillate.
The trick is, an amplifier might oscillate in the low MHz, and combined with stray capacitances of nF to uF, we can end up with an accidental feedback network that makes the power supply oscillate. This is plausible in breadboarding situations. This is the most likely failure condition, I think.
In lieu of a complete analysis, the best preventative against this situation, is probably a nice little electrolytic capacitor, say 1-10uF, across the supplies. This is better than a film or ceramic cap of the same value, because the electrolytic has adequate ESR to dampen oscillations in most cases. Additional bypass (usually local to other chips) can be, say, 0.1uF ceramic or whatnot.
Tim