To remove a palm tree root, it helps to understand how they grow. Many species of palm have extensive, fibrous, deep root systems, especially as they mature. Some, like the sabal palm, can go as deep and wide as 3.6 m (12 feet). Once you've removed the canopy and trunk of the palm, there will still be a lot to do to eliminate an unwanted or dead palm. Remove as much of the stump and roots as possible, leaving the rest to decline and die over time.
Sink the point of a shovel into the ground, 45 cm (18 inches) from the root crown of the palm tree stump. The root crown is easy to see, as it's usually a half-round shape, sitting slightly above the soil line.
Cut through fibrous side roots as you dig in a circle around the stump, prying up as you go. If you meet too much resistance, move the digging perimeter out another 30 cm (12 inches) to create a trench.
Dig down until you are under the main root ball of the palm. Pry up and pull out the stump. You may need assistance from another person or several people to do this, depending on the size of the palm root ball.
Backfill the hole where the root ball sat with the soil you dug out. Add more topsoil where the root ball displaced it. Tamp it down firmly with your foot. Any fibrous roots that remain will slowly decay in the soil.
Chop up the palm stump and root ball with an axe and use for compost or mulch, or dispose of it in garden waste bags.
Palm trees smaller than 1.5 m (5 feet) tall can be transplanted by careful digging of the root ball and the entire tree. Leave 1.2 m (4 feet) of trunk on the palm tree an use it as a lever, pushing down on the remaining trunk to lift up the root ball.
Don't try to lift a large palm tree root system by yourself. The thin, fibrous roots will have a lot of heavy soil clinging to them.