I worked on a Strat-style guitar recently. It was a HSS configuration and there was no output from either of the single coil pickups.
I opened it up to start troubleshooting and spotted the (slightly odd) problem pretty quickly.
But before I get to that, a short primer on single coil pickups.
Single Coil Pickup Construction
Most single coil pickups are made along similar lines to the diagram below.
The important things are the coil of wire and the pole pieces. These are what make a pickup work.
On a Fender-style single coil, you'll usually have six pole pieces with the wire wrapped around them a few thousand times to form a coil. All six pole pieces are magnetised, so each functions as a small magnet. Thanks to the magic of electromagnetic induction, a string vibrating in a pole piece's magnetic field will induce a current in the wire coil. This current goes to an amp and comes out as rock and roll (all going well).
Alternative pole piece magnets
Now, there is another option. Instead of magnetising all of the pole pieces so the each become individual magnets, a pickup maker can just attach a bar magnet along the bottom of all six poles. Because the magnet touches the poles, it extends its magnetic field through them, effectively magnetising them. The end result is a little different, tone-wise, but the principle is pretty similar.
Back to the troubleshooting
Back in the Strat with the unresponsive pickups, I found this:
The wiring's all ok and, at first glance you might think the pickups are looking ok. Wait, though, what's that rectangular 'ghosting' on the bottom of the single coils?
And what are those black bars stuck to the bottom of that humbucker?
Well, would you look at that. A pair of bar magnets.
Without these attached, the single coils will have no magnetic field, and therefore no output*. Whatever adhesive was originally used to attach the magnets perished and failed over the years. The magnets fell off and made their way to couple themselves to the magnet in the humbucker.
The fix is easy now.
A little clean-up to remove the worst of the old adhesive, a little poking about to figure out the proper polarity for each pickup and, and I've epoxied these magnets back in their homes.
*I suspect that, if I'd turned my amp way up, I might have gotten a very weak output from the pickups without the magnets attached. It's likely that the pole pieces would have retained a little magnetism. Not enough for much output but possibly a little.