Was having the same problems on my bike. Press the starter button and nothin happened. I actually took the solenoid apart to clean it at one stage. Problem kept happening.
I ended up replacing the relay under the tank but didn't think it could possibly be the problem and guess what it was!!
Havent had it happen since. I payed about five bucks for the replacement its worth the bucks just to see if it is the culprit.

Stewart Chrome Accessories for 'Modern Classic' Triumphs