This is a quick project and great skew or straight chisel practice. It will also use up all those scraps from woodworking and wood turning projects. To make an icicle you need straight grained wood blanks about 5 1/2" long by 1/2" square. Aside from a small skew or some small spindle gouges, you also need a four jaw chuck with small jaws.
With the blank held lightly in the chuck, bring up the tail stock and center the point on the blank end. I am using my One Way chuck with the small jaws.
With the blank aligned, tighten the chuck and snug up the tail stock. This gives a bit more support for the initial turning. It is not necessary but nice.
Rough down the blank and taper it toward the tail stock at the same time. I like a roughing gouge but at this size a skew does a nice job as well.
From here on in I like to do most of the wood turning with a 1/2" straight chisel that I reground from a Dollar Store carpenter's chisel. It is ground straight across and then beveled on both sides like a skew. I find it works at a more comforable angle than a skew and the carbon steel takes a sharper edge than does high speed steel although it does not hold it as long. The other tool I like to use, especially around the head near the chuck is a 1/4" round skew chisel made from a high speed steel tool bit.
Continue to taper the blank freeing it from the tail stock as you go. From this point on it is necessary to support the blank with one hand while turning with the other. After a while it is second nature.