In addition to finding pointed dripstone throughout caverns, you will also be able to grow your own.

If you have a stalactite hanging from a dripstone block that has a water source above, it will slowly grow over time. Additionally, a stalagmite will grow directly under the hanging stalactite if there is a solid blow within eleven blocks below. In the image above, you can see dripstone growing from the top, as well as forming on the bottom.

This is a great way to get more dripstone. To collect, simply take a pickaxe to the lower row of pointed dripstone.

Overall, dripstone takes a long time to grow. Items in Minecraft grow according to 'ticks'; an in-game day lasts about 20 minutes, which is 24,000 ticks. Dripstone has less than a .1 percent chance to grow each tick, so this means that it will take a while to grow dripstone. With several stalactites set up though, you should gather a decent amount of pointed dripstone in no time.