Spring noise! I HATE it with a passion. Here is a solution!
Kitchen Roll/Toilet Roll
In the past I've resorted to shoving kitchen roll beneath the springs which isn't really too desirable as when you use the tremolo it gets shredded and goes everywhere. However it does completely deaden the ringing but at the expensive of high end which leaves you with a slightly muffled sound which I cannot stand.
Floyd Upgrades Noiseless Springs
These looked ideal as they claim not to affect the tone but completely remove the spring ring.
I haven't tried these personally but I have a few friends who have who I trust a lot when it comes to tone and general tech knowledge.
They all had the exact same issue with these. The noise was gone but and this is a BIG but. It caused a grounding issue due to the coating which is potentially worse than the ringing IMHO.
Now onto the SSB Audio Noiseless Springs.
I got these on the off chance just because my new parts guitar had the worst spring ring that I have ever encountered in my life. I installed these today and recorded some before and after clips and it completely eradicated the noise and didn't alter the tone.
I must say that I am very happy and will be replacing the springs in my other trem loaded guitars ASAP with these.
Here is a link to purchase them if you feel like they could work out for you as they worked amazingly from me as you can see from the following clips.
SSB Audio Noiseless Tremolo Springs