Palm trees grow a root system that is completely different than other types of trees. Other trees have multiple growing points and have a vast woody root system that spreads laterally in the ground. These roots spread and grow from various points along the entire root system. Palm trees have a "root initiation zone" and grow roots from only this location. It is located just underneath the soil line. Palm roots are easy to dig out, so it is a relatively simple process to remove a palm tree stump.
Dig a trench around the stump of the palm tree. Dig the trench 6 to 12 inches away from the trunk and 1 foot deep.
Dig under the trunk of the palm. Push the shovel through the roots of the palm to sever them. You do not have to remove the entire root system of the palm; the roots will decompose and the palm cannot regrow from the root system.
Lift the stump of the palm up out of the ground. Sever any remaining roots and discard the stump.
Fill in the hole in with soil. The site will be suitable for immediate replanting.