I did think the fuel pump is a bit sticky of my '92 Cooper. Sometimes it energises when I turn the ignition lights on. Sometimes it doesn't. She runs well once she is has been persuaded (Pumping the throttle, turning the key and cursing at her and trying to stall her in to ticking over)
It was just getting her started that is the problem.
Until yesterday...She had a quarter of a tank in her, started kangarooing then cut out on me. Would start AT all.
The fuse for the indicators and headlights blew out too!!
Check the fuses behind the air filter casing, as one of this is for the fuel pump, (brown wire with a slate grey tracer), and are very prone to bad connection problems.
If your car still has the original white plastic twist and lock inline fuseholder it would be best to cut this out and solder in a blade fuse type holder with a waterproof cover.
In addition there is another fuse with white wires, which is for the ignition system and it would be a good idea to replace this one at the same time.
Other possibilities are a dodgy fuel cut off inertia switch, main relay pack, (situated on top of servo), which contains the fuel pump relay, or other wiring problem.