I use a combination of hand techniques which I find appropriate for doing different things on my drum set.
Firstly, I would say my basic snare hand technique is a matched grip following the G.L. Stone school as taught to me by my teacher Chuck Brown who studied with Stone.
The principles are to hold the stick between the thumb and first joint of the first finger (for light playing) or thumb and second joint of the first finger (for heavy playing), creating a balance or fulcrum point from which the stick is free to pivot.
Then the second, third, and fourth fingers are slightly curved around the stick and line up down the shaft towards the butt end of the stick. The wrist is positioned palm down so that the tip of the stick, the thumb/finger pivot point, the wrist, elbow, and shoulder joint are all in a straight line when the upper arm is allowed to hang naturally at the side.
In the up position, the wrist is drawn back, and the fingers in (closed), so the stick points straight up. In the down position, as you strike, the 2nd, 3rd, and 4th finger are relaxed and extended (opened) to release upon impact, allowing the stick to rebound freely and eliminate any shock to the wrist or fingers. The wrist and fingers then “get behind” the rebound and lift and close into the up position.
This pulls the sound out of the drum or cymbal by setting it in vibration and allows it to continue vibrating by getting off of it as quickly as possible. If you don’t do this you A) choke the sound, and B) damage your tendons, joints, and ligaments.
Now, if multiple strokes are to be played, the wrist stays down on the first stroke and the fingers remain open and play the next strokes by finger bounce - taking the energy from the wrist stroke and catching the rebound, then by slightly closing the fingers for each additional stroke to literally “dribble” the stick with the fingers.
This uses the smaller muscles of the fingers instead of the larger muscles of the arm and allows less effort to be put out when playing long, fast single stroke rolls.