Most specifically rust. BXs seem well protected in relation to other cars of the same era, but look out for inner wings, A posts around the door hinges (they can also crack in this area), bottom of the C post, sills, and boot floor (especially on the hatchback).
Rear suspension arm bearings - if there is creaking from the rear as the suspension moves, suspect the rear arm bearings. Also look out for the rear wheels leaning inwards at the top, like this. / \
The rear suspension struts can corrode though and cause a high pressure LHM leak, especially on estates where the strut walls are thinner due to the larger piston diameter.
Head gasket failure - diesels seem more prone to this than petrols. Probably caused in the main down to poor maintenance (dirty coolant/deteriorating radiator) and most models having no temperature gauge, so by the time the warning light comes on it's usually too late! Symptoms include rapid pressurisation of the cooling system from cold (listen for a hiss say 30 seconds after a cold start), 'hard' coolant hoses, oil in coolant, water in oil, unexplained water loss, steam from the exhaust etc.
Small LHM leaks from around the base of the front suspension struts are likely to be the strut return pipes - cheap and easy to replace but worth going for OE pipes if possible.
A hard ride is often down to flat spheres, although less common now as more BXs are enthusiast owned. Fairly easy to change, at least at the front and can be got for around £20 each. A short 'tick time' (every few seconds) can be down to a flat accumulator sphere (cheap and easy to replace) or a worn hydraulic pump (not too bad to change but pricier). Problems with the latter will often be accompanied by intermittent loss of power assistance from the steering.
Pictures welcome of any trouble spots for rust etc, I'll try and add a few as time goes on. Also feel free to add any more pointers below - ideally it will all be condensed into an easy to read guide at a later date...
Below is a guide from Car Mechanics magazine, originally scanned in by DLM.




