I had a problem with my da9 integra doing this exact thing cutting out all of a sudden, randomly(usually at an intersection....) cranking without firing, then it would restart 5mins later it turned out to be 2 causes:

First one the car would cut out and the check engine light would come on, car would crank but would not fire, checked the ecu and it threw an ignitor code, so i replaced the igniter and the coil. The car was fine for a few hundred k's

The second cause i realised that there was no power to the dash when turning the ignition on, also was no power to the ecu, no main relay click. I had already checked the main relay's solder joints and they were fine but had a spair one in the car, i changed it and still nothing. This turned out to be the ignition switch which is a common fault on honda's (wish i had known this, it cost me a $330 tow back to wollongong)

Hope this helps anyone with a intermittant no start/ cut out issue. and hopefully this is the last of mine.