Left hand release can add a lot of character and style to your phrasing. It's a way of controlling your sound with your fretting hand. Releasing your fretting hand at the right time gives the mandolin chop and banjo vamp its percussive effect, it can also help with separation of notes on banjo.
A good way to develop control for this skill is by starting with a single fretted note. Set a metronome to 60 BPM. Hold the note down and pick it. Let it ring for 4 beats. Do this several times to lock in with your timing. Now, pick on beat 1, and release your finger pressure precisely on beat 3. Don't remove the finger from the string, it should remain in contact, but relieve enough pressure so the note deadens. This it the coordination we are going for. Continue to do this several times until comfortable and certain your pick attack and left fret hand release are syncing up with the metronome. After that, pick on beat one, then release on beat 2. Then begin to increase the metronome by 10 BPM at a time and repeat the exercise until comfortable. This is demonstrated in the video below
Eventually you want to be able to let off just after, or sometimes even just as you play a note or chord. Sometimes, to get an extremely percussive effect, you don't actually push the string down far enough to contact the fret.
For muting open strings you can let a finger drop on the string to deaden sound. Usually the pinky and ring, but it can be any or all fingers. Doing this well and in time is a key component of Sam Bush's chop, and is a great technique to add to your arsenal. The exercise is the same as above except you drop your finger on an open string to mute it with the beat.
Try and listen for this when your favorite record is on. Take notice of when a player plays a note short and leaves space instead of a ringing note. Then, experiment with playing some notes short and some long. Find the combination you like. The possibilities are endless.
This can be used in your solos, and rhythm, on any instrument and in any style-especially fretted instruments. Watch the video that demonstrates the B part of Turkey in the Straw. It's an example of how a melody can be played by giving the notes full value, or using left hand release to create space and add some character.