I recently bought a black, unbranded Flying V copy for £70 and it sounds pretty damn amazing, surprisingly...

However, there's a slight problem with the bridge pickup; as soon as I crank the gain on my amp (a 1987 JCM800 100w Combo...) there's a strange feedback/whistling sound which goes up and down in pitch. If I move 7/8 feet away from the amp, it stops but this is only at bedroom level. I suspect that at gig level, it'll be harder to rectify...

I have a Gibson Les Paul and at the same settings, everything is absolutely fine as it is with every other guitar I own. I have no idea what is causing it, but I think it's fairly obvious it's a problem is with the pickup itself. Thing is, apart from the whistling, it actually sounds very nice so I don't really want to change it if I can avoid it...so any suggestions would be welcome!

I hope this description is helpful is one way or another...if people need demonstration of the prob, I could make a video in the next few days!

maybe the pickups have gone microphonic? check the guitar's shielding and the ground wire.
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Afterwards, Han Solo said "Great shot kid, that was one in a million!"
The pick ups are probably cheap. You can swap them out for better ones or have them potted and shielded.

you can try potting them in melted wax, i'm sure there is a howto inthe gb&c here on ug. otherwise, get a pair of new pickups, but give a try to the wax, if you screw something well, 2 cheap pickups lost (i dont know how you can screw something that way but, oh, way of screwing thing are endless )
Peavey Bandit 112
Washburn Chicago series :SD Invader at bridge ,GFS Lil Killer on neck
Bc Rich Warlock :GFS Vintage59 on neck ,Crunchy rails on bridge
Epiphone SG :Tusq Nut, GFS Crunchy Pat on neck, bc rich pickup on bridge
lots of cheap pedals
Thanks for all the answers guys - gonna pot them this week. I'll let you know if all goes to plan!