Muting, for me, is a two-handed technique. Allow me to explain:
Let's say I'm playing a line on the A string. On my left hand, the tips of my fingers fret the string. Now, assuming I'm playing a single-note line, there are three fingers at any one time that aren't fretting a note. I use these fingers mainly to mute the strings above the one I'm playing. Playing the A string with my index finger? D and G strings are muted by the upper parts (not the tips) of my other three fingers. My middle finger, being longer than the others, is sometimes used to mute strings below the one I'm playing. My left hand is very flat to the fretboard as I play. I like to keep as much of my hand in contact with the strings to mute as possible.
Pictures!
Step one: C#, fourth fret, A string. Single note held down, no muting. If I flail at the strings I'll sound E C# D G, which is a lovely C#mb5addb9 chord (ew) but not what we're looking for.
Step two: mute those pesky D and G strings with my little finger. That'll learn 'em.
Step three: don't want that low E ringing out either. Mute that with my middle and ring fingers (ring finger tends to follow little or middle fingers. so long as it's not muting the A string we're golden).
Now, the right hand can flail away at the strings all I want, but the only clear note that'll come through is the C# that I was looking for in the first place.
On the right hand, my thumb is nearly always used to mute the strings below the one I'm playing, except of course when I'm using my thumb to sound the strings (slap, thumbing, or picking). If I'm playing the bottom string, my thumb rests on the pickup, pointing straight at the ground. If I'm playing any other strings, the thumb rests on the strings below. If I'm playing the G string (or C on my 6-stringer), the thumb covers all other strings. Between left hand fingers and right hand thumb, I've got the other three (or four, or five) strings on lockdown!
It's all about economy of motion, in my opinion. When I play a note, I don't just concentrate on that note. I anticipate the next note, and this affects my choice of finger (if the line goes Ab > F I'm not going to play the Ab with my index finger). I also look at the notes I'm
not playing, and mute them accordingly, making sure that I'm not making things difficult for myself.
Man it's difficult to explain finger positions in text. Hope I helped!