The majority of engine fires occur at start-up. Most are the result of guys using the carburetor to "prime". If you take an average carburetor off an airplane and actuate the throttle you'll shoot a pressurized stream of raw fuel straight up from the carb throat. That's the aircraft equivalent of an accelerator pump. When guys pump the throttle to start an airplane they use that pump to shoot fuel into the induction. If the engine starts it creates a significant draft and all that fuel is drawn into the engine. If the engine doesn't start right up some of that raw fuel will fall back through the carb, into the airbox, and drip into the engine cowl. Not a good place for fuel and it can definitely catch fire if an ignition source is added. The most common ignition source is an engine backfire and those are usually caused by a lean condition, whether because of induction leak or poor fuel vaporization after pumping the throttle. If you have fuel drop out and a backfire and the fuel ignites your best action is to continue cranking and hope the engine starts and sucks the burning fuel into the engine. If it fails to start your best action is to grab a fire extinguisher. Most of us that live in cold climates have primer lines into all cylinders. I can pump my panel mounted primer as much as I want and fuel will not drain into the induction and into the airbox. Using the primer system is much, much safer than pumping the throttle. If you must use throttle to prime make sure you turn the key and start the induction draft before pumping the throttle. Never pump throttle on an engine that isn't spinning.
I had an induction leak lean cylinder, was frustrated that the engine wouldn't quite light up, and pumped the throttle to help it. The resulting induction fire was extinguished quickly. It took me about 5 seconds to unstrap, slide the seat, get out, grab the extinguisher, etc. The engine compartment damage was about a thousand dollars a second. Without the extinguisher the plane would have burned to the ground.