Sounds like a starter problem to me as well, but.
The solenoid dropping out is a bit mysterious.
1. When you press the start button, I take it the lights go out. This should happen as the starter relay causes the load relief relay to de-energise thus putting the lights out.
Alf: Don't know 100% but will check tomorrow. Fairly sure the lights go out.
2. Does the starter motor sound like it's got full power to start with, or does it just turn slowly as if the battery was flat.
Alf: Full power to start with for maybe 1 or 2 turns then a whirling noise and no engagement.
3. Are you getting anything unusual like the clock resetting when you press the start button. This is an indication you have a short across some of the starter motor windings, and the starter motor is effectively dropping all your battery voltage across it.
Alf: Very interesting - clock has been resetting itself for the last few years despite battery changes. Clock starts when ignition is turned on. I have a stopwatch rather than a clock.
4. If the starter motor turns over normally, and then the solenoid drops out, it could be that your starter relay ( big yellow one ) is dropping the voltage across the contacts ( carbon build up leading to high resistance ) and this is causing your starter solenoid to drop out. One way to check is to apply 12v directly to the terminal on the starter motor that the black lead attaches to. Try and leave the black lead attached if you can otherwise when you do this the load relief relay will not de-energise.
Alf: Too complex for me but I'm meeting up with some mates tomorrow who are far more mechanically savvy - I'll ask them to explain!
5. If you want to test the starter motor off the bike then it's negative to the case. Positive to the big terminal. And then apply a positive to the small spade terminal (where the black lead goes ) make sure you have the starter physically secured when you do this. If you don't you'll understand why.
Alf: I get the idea - thanks for the heads up. Reminds me of the time I decided to take the clutch slave cylinder apart and almost took my eye out.
Can't think of anything else you could check short of replacement or dismantling to examine.
Ian