Progress!
I checked the output cap using an Ohmmeter as suggested, and it appeared to be ok, reading a rising resistance. It topped out around 11k, but I guess the actual reading is kind of meaningless, isn't it?
Then I tried a 470uF cap in parallel with the output cap (80uF). Bingo! With just that capacitor as load, the scope reads a dead flat line with no noise or ripple visible.
Testing with a resistor in parallel, things looked ok but as I turned up the supply beyond about 10V it began current limiting and then blew the top off the capacitor, which I guess was a bit underspecified at 35V
Would things have been better if I had turned up the voltage more slowly?
Brushing aside my poor test procedure (ahem! beginners' forum!) does this mean that the output cap needs replacing? I suppose it is still working as a capacitor but no longer the right value to keep the circuit stable. If so, what should I look for in a replacement? Clearly it should be 8uF 300V as specified for the original, or at least close to that, but is there any other spec to bear in mind or any recommended brand or series for such parts?
Many many thanks to you all for getting me this far. I think we are almost there -- let me know if you disagree!