I have a 1998 Honda Accord that intermittently won’t start. When the issue occurs the dashboard lights, radio, etc all work just fine, but the engine won’t turn over (no clicking, no noise at all). It’s standard vehicle, but I definitely have my foot all the way to the floor to depress the clutch safety switch. The issue only seems to occur when the car is warm (i.e. it never happens on the first start in the morning), although it’s been known to happen even after 2hrs of cooling down. The issue used to only occur 1 out of 100 starts, but now it’s getting closer to 1 out of 4 times. I’m able to get the car going by jump starting it, but that’s getting old. 
Any clues?
The problem might be with the clutch pedal interlock switch. As a quick test you can unplug the switch and then insert a jumper wire into the plug to remove the switch from the circuit. If the engine now starts everytime replace the clutch pedal interlock switch as this is a safety feature.
Tester
In the early to mid 90’s the Hondas had a common problem with the starter relay or the main relay. I had a 91 with the same issue. Doesn’t do anything randomly when you try to start. Happens more in hot weather. Jumping seems to work even though the battery isn’t dead mostly because the higher voltage from the running car can get the relay to activate. Test the battery first and make sure it isn’t on it’s last legs. Then the relay would be my next guess.
This model is also subject to an ignition switch Recall which can cause various types of problems.
If the repair has never been performed under the Recall you might consider having the Honda dealer do this for you. It’s free of charge as to parts and labor.
There are several other Recalls out on this model so I’d get them all done in one shot if none of them have been done previously.
That main relay is the FI relay, I think. You’re talking about the relay that needs resoldering when it breaks? If so, the starter will still turn over.
Thanks for all the info! I’ve jumpered the clutch interlock switch and will let you know how it goes. For those experiencing the same issue, the Accord has two clutch switches (see attached pic - presumably both can fail?). I’ve started with the interlock switch.