The ice pick is a fair secondary backup weapon (or even primary if you prefer stab-type knive weaponary). It is light, and in terms of durability, is better than the kitchen knife but worse than the hunting knife. As the Ice Pick is one of the weapons with the 'Jaw Stab' attack, it can kill a zombie instantly if close enough and timed correctly.
The Ice Pick is a single attack weapon, so it is best if used in single combat or if zombies are relatively spread out when attacking.
Its tree/door damage is negligible. (2 tree, 1 door per attack).