My 1985 K100RS has me worried.  Last weekend she displayed a reluctance to start (in the past she just sprang to life at the touch of the magic button).  Today she flat refused.  I avoided a long walk home though, as I managed to bump start the heavy beast. 
I bought her with 22K miles on the clock; now there are 40K.  I've decided to pull the starter tomorrow and give it a good cleaning, as I've never done that.  But will that resolve the issue? I've read a bit about carbon build-up interfering with the earth for the electrical system.  However, I thought in such cases the lights would also be dark.  When the bike refused to start today the lights in the instruments and the headlight were on.  But pushing the start button resulted in absolutely nothing -- no click, no burp, no whirr of the fuel pump.  The battery is quite new.  The voltmeter assures me it's healthy, and that proved true when I returned to the garage and the bike started immediately, as if nothing out of the ordinary had happened. 
Tomorrow I plan to troubleshoot the problem.  While I'm in there cleaning the starter what else should I do? Would this possibly be a relay issue? If the starter relay is operational, should I detect an audible sound when the button is pushed to start the engine?
Thanks in advance for your thoughts.

Just as a quick check, look at the four-pin connector going up the right hand side rear corner of the fuel tank - it's got the fuel pump/fuel level wires in it. They often get grimy and lose good connection. The wires are brown, yellow, white and gree....I can't remember. My K100RS is too far away in OZ. And of course, there's the fuses and their connection. I had a side of the road event some years back that started out as intermittent but ended up stopping me flat. It turned out to be was the fuel pump fuse connection wasn't tight.

The starter brushes going South is often accompanied by a chorus of blinking lights; left to go right, horn to go left, high beam flashing as you ride along, etc. Like a ghost in the machine. It's pretty cool, actually (if you know that's what's happening when it happens). Otherwise it's scary.

I am assuming that when you push the magic button the starter does not spin.

I am not familiar with your particular year, but I am pretty sure that there are some safety interlocks in the starter circuit that could prevent starter operation.  I would check them before anything else.

First off, does the neutral indicator light up?  What happens when you pull in the clutch?  If the neutral isn't working properly, pulling the clutch should bypass it.  It's possible, but unlikely that both are bad.

Do you know if your bike has a side stand switch?  I believe there are some models that have that switch, and if it fails, or is out of adjustment it can prevent starter operation.

Cleaning the starter is a good first place to go when the bike won't start. I do it a couple times a season as a precautionary measure now days.

brickrider wrote:....
When the bike refused to start today the lights in the instruments and the headlight were on.  But pushing the start button resulted in absolutely nothing -- no click, no burp, no whirr of the fuel pump.  The battery is quite new.  The voltmeter assures me it's healthy, and that proved true when I returned to the garage and the bike started immediately, as if nothing out of the ordinary had happened. 
....

"no click, no burp, no whirr" >> the starter relay coil is not being energised. You would hear the relay click if nothing else.
"instruments and the headlight were on" >> ignition sw is working. Fuse #1 ok
The fact that it started immediately back at the garage indicates its intermittant... like a loose or corroded connection??

So it must be something inbetween, like neutral light, clutch sw, starter/kill sw .75
Clutch sw has a connector under the tank, front/left I think.

If there is no click of the starter relay or whir of the pump then my money is on the starter/kill sw. Its a normally open switch and if it gets corroded there's no continuity. Probably just needs a good clean. Would explain the intermittant nature also.

OR it could be stuck contacts in the starter relay (probably not tho if it started ok back home)

Also, the starter relay coil is grounded thru the ICU. Wouldn't be the first time we've seen a loose connector there!

I'd also make sure all your ground connections are very clean and deoxidised. GB, starter, under tank.

Update: I pulled and dismantled the starter.  I found less carbon dust inside than I expected.  She's 30 years old, but has a mere 40K miles on the clock.  Truth to tell, I don't have any reference point to judge what is a lot or a little carbon dust.  I'd never before dismantled a starter. I didn't find any corrosion on any connectors that looked alarming, but there were a couple that were not as snug as I'd have liked.  Bottom line at this juncture is she preformed splendidly on reassembly and a test ride.  She started immediately a number of times, so I'm going to be optimistic and believe the murky issue(s) have been resolved and she is ready for some summer fun.
Thanks to all for suggestions of where to look/what to do.

The starter has been my go to everytime for starting issues. That was probably the first quirk I learned about with these bikes.