You definetely need to incorporate both types of muting to get a nice clean playing style. One thing that helped me a lot was using my fret hand first finger to mute the strings above and below the one I'm playing. Its a little hard to explain. basically you fret the note the same way just kinda extend your first finger a little higher so it's touching the string above it and gently resting on the string below it thus muting them both. For the strings ur not playing you can use your picking hand to mute those out. So if you were playing on the 2nd string...your 1st and 3rd string would be muted by your 1st finger...or other finger depending on the lick...and the top 3 strings would be muted by your fret hand.

Hope that helps a little.