Sometimes I want a clean cutoff at the end of a note. The common method is to release the pressure applied by the fretting finger. You don't want to take it completely off the string, as the open note will sound, like a pull-off. Just enough to unfret the note with the left hand still muting the strings.
That was my technique for a long time. Then I got tired of that tiny bit of fret-buzz that occurs in the process of removing the fretting finger. There's a moment when the note is still sounding and the finger isn't applying enough pressure to keep it firmly fretted, so it buzzes out on its way to being muted. I added a picking hand mute with the side of my palm in conjunction with the left hand muting. This is closer to the sound I want, but my right hand isn't in the best position to immediately resume playing if it's only a brief pause.
I discovered that I can use my free fingers on my picking hand to mute individual strings when want to stop a note. My ring finger feels most natural. I can strike a specific string very quickly with the flesh of my ring finger, and the note is instantly dead, no fret noise. The damping is more rapid than with the side of my palm, so the sound is even better. And I don't have to change my hand position, so I can stop and start playing with ease.