Intersting - I have a tremol-no installed, thought that could be a source of this - have tried disengaging it, but not removing it. Interesting to hear you've had that same problem before installing the tremol-no .
I had to remove my Tremol-No because I couldn't stand the slight rattling of its screws when it wasn't completely locked. Drove me nuts...
Put a capo on, does it still do it? If so it likely not nut upwards BUT I have heard very similar on several LP's - worn nut slots every time. Move truss setting slightly also just to kill that bird 
Use a little Nut Sauce on nut and saddles and trem pivot points. It's always easier to test if you have someone play while you press on various points too.
After doing all tests again with two people - have a look at the electrics. I'd firstly resolder every joint and retry, if still there, personally I'd look at the output on an o'scope, disconnect the pickups one at a time, put them direct on the scope and play, compare, rules out rest of electrics if still there. If you have't a scope - take it to a tech. Or to rule out the pickups and electrics hold a known good pickup, connected to the scope (or amp) direct over the strings, get your buddy to play and see if that picks it up.
Was there ever a solution to this, I've had this for many years (maybe 15 years when first noticed with the Cybertwin) and it seems to be more noticeable on digital equipment (cybertwin, GSP1101 and Kemper) I don't notice it nearly as much or if at all when I play through my Fender Deluxe with a TS9. I've been using a tweaked profile of "teeny tiny feet" with slight distortion for an SRV Pride and Joy type of sound. I just replace my pickups with Lindy Fralins and it seems to have come back with a vengeance! I would describe it as a very short "hitting a metal trash can lid with a rubber mallet" type sound just when the string is picked. I wish the link for the example still worked so I could compare it. I believe I got around it before by lowering the pickup WAY low, and will try it again tonight but found this thread while looking for a remedy and just thought I'd ask. I also have a Music Man JP6 and that sounds excellent thought the same profile.
Was there ever a solution to this, I've had this for many years (maybe 15 years when first noticed with the Cybertwin) and it seems to be more noticeable on digital equipment (cybertwin, GSP1101 and Kemper) I don't notice it nearly as much or if at all when I play through my Fender Deluxe with a TS9. I've been using a tweaked profile of "teeny tiny feet" with slight distortion for an SRV Pride and Joy type of sound. I just replace my pickups with Lindy Fralins and it seems to have come back with a vengeance! I would describe it as a very short "hitting a metal trash can lid with a rubber mallet" type sound just when the string is picked. I wish the link for the example still worked so I could compare it. I believe I got around it before by lowering the pickup WAY low, and will try it again tonight but found this thread while looking for a remedy and just thought I'd ask. I also have a Music Man JP6 and that sounds excellent thought the same profile.
I took those files down literally within this week, sorry man
Nah, I never found the solution. I don't think there is one. Maybe it's just part of the sound. I decided to stop worrying about it.
If you have a strat style guitar, try muffling the springs on the trem as I had one that created terrible metallic overtones when I played.
edit: read the initial post and see where they tried the trem springs.
[Blocked Image: http://i976.photobucket.com/albums/ae247/aflory2000/stuff/fire/fire1.gif]Honestly, after all that, this is where that puppy would land!!! lol
Have you tried plugging into a Good tube amp, instead of all digital????
Just a little update, it's pretty much the whole guitar, even open strings. I've lowered the pickups probably double what they recommend, 1/4 inch on low side and 3/16 on high side (i believe Fender spec is 1/8 and 3/32 from string when held down at last fret)
I did hold the springs with one hand and picked open strings with the other but it didn't make a difference.
I performed all the adjustments (action, intonation and pickup height) originally on my Fender Deluxe sitting in an upstairs bedroom because of better light. Kemper and rest of gear is set up in basement where its a little too dark for guitar maintenance.
Tomorrow I will run it through a TS9 and my Fender Deluxe Reverb and put foam in the springs
It's definitely related to the amount of high end your pickups produce. The middle and bridge pickups on my CS strat exhibit this behaviour with certain profiles, more often than not they're profiles created with humbuckers in mind. Try reducing definition in the Amp module or try profiles that were made for single coils. For me, this is where the likes of The Amp Factory and Bert Meulendijk really shine.
sam, you mentioned swapping a few strings but never changing them all. I would certainly try that. It's obvious something is vibrating in a strange manner or being excited by your playing causing a strange resonant frequency. I have seen strings cause all sorts of oddities. They can have uneven thickness causing strange vibration patterns. The resonance of the guitar can be exciting the other strings in a strange way so that even when muted somewhat produce some type of dissonant harmonic. Anyway, a cheap test
