"singing" Tube Tone?

Hey Guys,

I've had a question for quite a long time and I hope it is okay that I've opened a new topic. I play for like 4 years and I'm kind of unhappy with the overdriven lead-sound of my gear.
I really would like to create that overdriven "singing" sound you hear very often in distorted riffs. At the moment I'm playing a litte Satriani and I miss the high peaks of my distortion. Is there a special way to hit the string if you want to get it? I think it would be a great way to pronounce some notes better but I won't get the tone on purpose. Even when I go max-volume with my amp with an extra distortion pedal I only get bad noises at the end. I hope you know what sound I'm looking for. If not I could add some links of lessens where you hear it.

My gear:
Speaker: 4x12 Stramp Cabinet (not sure of the Watts)
Amp: Peavey Classic 30 Head
Guitar: Gibson Les Paul Standard 2008
Used Effects: Line 6 - M13

Best greetings from Germany!

Eike