Ohm's law says sqrt(P/R) = I. So your resistor can handle at least sqrt(0.25/100) = 50mA (not much).
In practice, because parts are designed to do exactly the opposite of what you want (e.g. not explode), they can handle much more than that (don't ever design in parts beyond spec in products or anything you ever share with anyone!!!).
10X that (>500mA) should do the trick, but that doesn't necessarily guarantee that it's internal ignition will result in external flame. Many parts are coated to prevented this and the chemistry is always being improved to further reduce the risk of a secondary ignition. That's how products catch fire (a VERY bad thing).
To get 500mA through 100 Ohms you need V = IR = 0.5 * 100 = 50V source. That's about 30 E91 ("AA") batteries in series.
If you reduce your resistor to an 1/8W 1 Ohm part, you can relax your battery requirements much more. Repeating the math...
sqrt(0.125/1)*1 = 350mV, something easily achieved in just a single "AA" battery. At 1 Ohm, the current demand will be 1.5A (approx) from an "AA" battery, so you need a battery with 1C rating > 2A for safety (so that the battery doesn't ignite too!).
I don't think this is necessarily a bad question. Since it's clear that the OP is doing something dangerous that others shouldn't attempt. Sometimes you have to experiment "on the edge" to truly grasp concepts... just sayin'
Pulse handling is a highly variable characteristic. You can often find this this information in manufacturers' handbooks, occasionally on the
datasheet. Some resistors are specially designed for pulse handling capability and that will be well characterized. Things like the trimming method can affect the maximums- I would suggest not taking chances if reliability is important.
Here is such a resistor (Panasonic ERJ-P08J101V surge-resisting) which compares the pulse handling ability with more conventional resistors (note the test limits- 1,000 cycles and +/-5% permissible resistance change).

In this case, a 2/3W resistor can withstand 20-40W for 1msec (depending on type) so your 10x for 1usec seems fairly conservative (at least for the type of resistor I looked at, and with the allowable number of pulses and resistance change).
Best Answer
DO NOT destructively test resistors or any other devices without the DIRECT SUPERVISION OF AN ADULT QUALIFIED TO INSURE SAFETY
It depends on the wattage rating of the resistor, airflow, atmospheric pressure and composition, what the leads are connected to internally and externally, the temperature coefficient of the resistor, and the degree to which its materials are flame retardant.
DO NOT destructively test resistors or any other devices without the DIRECT SUPERVISION OF AN ADULT QUALIFIED TO INSURE SAFETY
You can find out if you are exceeding the manufacturer's rating by comparing the square of voltage, divided by resistance, to the rated power limit in watts. For example, 9*9/10 or 8.1 watts is drastically more than the 1/4 watt rating of most small through hole resistors so it's clear that the component is being abused and there is a substantial risk that something bad and potentially dangerous will happen. (1/2 and 1/8 watt units are also common, though you can get higher rated ones).
DO NOT destructively test resistors or any other devices without the DIRECT SUPERVISION OF AN ADULT QUALIFIED TO INSURE SAFETY
And don't assume the resistor will just burn; it could also explode. Either one can be hazardous to your health and surrounding property. Also consider that it could be hot enough ignite other materials or produce nasty gases even if the resistor itself does not burst into flame. You could also overheat whatever is sourcing power to the resistor.
DO NOT destructively test resistors or any other devices without the DIRECT SUPERVISION OF AN ADULT QUALIFIED TO INSURE SAFETY