Gears randomly shifting?

Thanks for the responses.

It seems that adjusting the L&H screws may not be the solution. The chain is not slipping off the freewheel - ie no over or undershifting is occuring. The gears are shifting correct.

The slipping happens in any gear so don't think it is worn teeth on the cassette.
Chain tension or arm tension sounds closer to the problem. When I turn the pedals the chain just doesn't have enough tension to take hold of the rear cassette.

How do I resolve this? Would I still need to adjust or reindex cables?

Clean and re-grease all the moving parts of the rear mech also.

Chain too long could also contribute perhaps? Would need to get hands on to really get to bottom of problem, but there's enough info now in this thread to help you get to the solution I reckon.

Hasn't he already messed with the limit screws? Might be worth putting those right before re-indexing.

Naturally, yes.