Before you start... Make sure you have a copy of the BMW E28 factory Electronic Troubleshooting Manual (ETM) for your model year E28. They are available here in .pdf format. Scroll down the page to the "Electronics" heading. Another ETM location has been found here, which has no relationship to being a BMW support site, so its the only BMW material present. Second, download a copy of my text listing of what electrical ground point is associated with what device, to help you troubleshoot ground fault problems. Without these two documents, you'll be stumbling around in the dark.

Stalling, won't start, or suspected fuel pump problem.
Make sure Fuse #1 is not blown, or has trouble making good contact due to corrosion or weak fuse holder arm retainers. Replace the fuse or check for continuity at the base of the fuse holder arms, not at the top where the process of checking may cause a bad connection with the contact arm retainers to make intermittent contact, causing you to think the fuse holder is OK, when it may not be.

If you now suspect the relay may be at fault, jumper pins 30 and 87, while listening for the fuel pump motor, or attempt to start the engine and see if it runs. The relay position shown below is for the 1982 to mid-1985 528e. It's location changed sometime after mid-1985 to the other side of the large black connector, between the relays. You should know that even if you can hear and feel the suspect relay "click" when activated, that is not an accurate indicator that it is fully functioning, since the contacts are what usually fail due to pitting and arcing and prevent enough power from being applied to the respective circuits. You must check with a voltmeter at the fuse or relay operated device, if you don't have a known working replacement relay to substitute. This also applies to swapping relays around, which may also be faulty and has certainly been known to happen.

If your relays are old I strongly suggest you replace them all, as I've seen many that are really troublesome on at least a half dozen different E28s, over the past year. I also suggest you always carry a few spare relays... certainly ones to replace the main, fuel pump and headlights.

With respect to keeping relay tabs and socket connectors clean, I suggest you purchase CAIG Labs DexoIT and SHIELD. I've been using CAIG Labs products since 1959, when it was known as Cramoline. You can find the products here. I apply DeoxIT SHIELD to all fuse and relay contacts once every year, which takes less than an hour. Every second year, I reapply both DeoxIT and SHIELD to all fuse, relay and grounding connections, as well as all device connector contacts, such as sensors and cable connectors. In the past 24 years, the vehicle has never had an electrical connection failure, other than a fuse or relay due to age, not due to electrical contact problems. Over 50% of all E28s that I have worked on have questionable electrical connections, so the problem is well known but not properly dealt with, in my experience with many E28s.

Battery Going Dead - If your battery isn't holding a charge, but is known to be in good condition and your charging circuit is working properly, look for something that might be draining the battery when the car is parked. Normal drain on the battery is usually between 20 and 25 milliamps, as measured on my car and several others that had no battery drain problem. If it is much higher on your car, something is consuming it, such as as stuck mirrior heater relay, bad antenna circuit, etc.