The clumping base of the fishtail palm rises into many slender gray trunks that end in wide, dark green and upright fronds. The leaflets lining the stems take the shape of fishtails. This palm likes well-drained soils and is moderately salt tolerant. 15 Gallon Fishtail Palm Trees-Miami. It grows well in partial sun but accepts shade and will flourish in the shadow of larger palm trees. The fishtail palm has a relatively shallow root structure. 7 Gallon Fishtail Palm-Broward. Plant it in sheltered areas or move potted plants indoors during windstorms.