A few days ago I removed the flooring under the middle seats in a Seneca II to troubleshoot a strobe problem. I found some grounding issues and fixed them. While doing multimeter checks for power at various pins and connectors, I began to see flashes from the wiring harness. Upon examination of the harness, I found the wiring insulation cracked in multiple places. The electricity was arcing through the insulation. It appeared intact at first glance...but clearly it wasn't.
The aircraft was having other electrical problems, and I'm sure this was at the heart of them.
I've had several electrical fires on board, including an air conditioning unit in a PZLM18 a few years ago which simply melted the switches and controls. I smelled smoke, then saw metal dripping out of the control box.
Another involved smoke from behind a circuit breaker panel in a Lear35. The cause was traced back to an improper voltage applied to some equipment installed on the airplane, which had smoked several components and fused several relays. I ended up on oxygen, terminating the flight (obviously), and returning to land.
In another case, again in a M18T, the improper setting of a hydraulic pressure switch allowed an electric pump to run excessively, ultimately fusing the relay that controlled it, and in the end setting fire to the pump. The cockpit began to fill with smoke as I passed over the numbers for an overhead break...and by the time I reached the threshold, the pump was on fire and the cockpit full of smoke from burning hydraulic fluid. When I reached midfield to exit the runway, I couldn't see out of the cockpit.
Most of these problems have been due to either a failure in the wiring (such as the arcing strobe power pack), or a stuck or fused relay associated with a high amperage item such as the hydraulic power pump. The learjet involved damage done on the ground, but not discovered until flight.
In the case of avionics, however, especially newer avionics with low power draw, new wiring, and continuous wire runs...you're not likely to see an increase in cockpit or wire fires.
Remember that circuit breakers and fuses are placed in a circuit not to protect the equipment in the circuit...but to protect the wiring. Under most circumstances you should see a circuit breaker pop, or a fuse blow or burn/melt, long before an electrical fire becomes an issue.
Remember also that in most cases an electrical fire is a matter of removing the electrical source. Once the ignition source is gone, the fire may go away entirely (often the case with burning insulation, for example), or may be easily put out. Not always the case, depending on location, but usually so.
Remember also that electrical fires are rare. Most often, even if wiring does become damaged, it will sever it's connection. Burn through the wiring, and that may be the end of that circuit. It may then become shorted and have a different path and a different problem, but if you can discover the circuit that's the cause, pulling a breaker or turning off a switch makes that problem go away for the most part.
Monitor anything powered by electricity. If you see a change in electrical load on an ammeter, question why this has occured. A sudden increase in load may mean a short somewhere, for example. It could be something stuck on and operating when it shouldn't be operating. A group of items or several items going dead at once, or all providing strange indications (intermittant operation, flashing, sluggish operation, etc) might indicate a short in a bus, wiring bundle, or harness...or cannon plug. These are clues that help you make decisions and take action. You may not see the problem itself, and you may not even smell smoke or see any firm indications of a fire...but these are all important clues which may help you address the problem before it ever becomes a fire.