Rolling is not the reemergence of rave culture; it's when you simply toss the ingredients from glass to tin and then back into the glass. This is a light touch and especially useful when pouring over ice and adding soda or additional water. Rolling is not to be used in the presence of heavy or hard to blend ingredients such as heavy cream, honey, or eggs.
Shaking is the most common method of rendering ingredients and takes on a multitude of forms. I personally use three shaking methods, all elevated over my shoulder alternating angles of the shaker while I rock it back and forth. I try to keep the pattern rhythmic so that I can hear the shards of ice within the shaker hit the walls.
1. First there's the light shake, which, while still vigorous, is short of a full 10 to 15 seconds of shaking by about 5 seconds. Part of the difficulty in offering either time or counts is that it depends a lot on your upper body strength--please keep this in mind if your results are varied. I recommend the light shake if you're straining something over ice, especially cracked ice. (Consequently the jury is out on what ice is best to use in the shaker, but the general consensus is to use lots of it.)
2. Next there's the full shake. This is 10 to 15 seconds of vigorous shaking. Use this method with hard to blend ingredients, such as honey or cream. Also, use this method when "volumunization" or dilution is needed.
3. The dry shake is useful for eggs. You begin by shaking ingredients without ice. Then add ice and proceed to the full shake.
Stirring is the most genteel way to blend ingredients. I use really only one method of stirring but at different rates. I place the spoon between my thumb and middle finger, much like chopsticks and use my pointer finger as a guide. You have to watch the water line and taste occasionally to check your dilution. This method works best when you have clear ingredients or all liquors.
Muddling is when you use a short, stout stick called a muddler to crush the ingredients. This is a common practice in such drinks as the Caiparhina or Mojito, although needn't be used exclusively as you can further the shake ingredients once muddled. The most important thing is to exercise restraint when muddling, as over-muddling seems to be an epidemic.
The swizzle stick, some of you may be surprised to hear, is an actual tool and not the logo-emblazoned plastic stirrers that are offered in hotels with your Gin & Tonic or coffee. The real swizzle stick is a rarity to find, but is used much like a primitive blender whereby you place the stick between the palm of your hands and roll it back and forth. The swizzle itself has multiple prongs that extend from the bottom. This stirs the drink while leaving ingredients at the bottom undisturbed.
Hopefully this will help illuminate these techniques for those of you who are just beginning and help clarify them for those of you who have already begun. I certainly think that there's always room to expand one's knowledge and improve one's own technique, so I welcome comments.