We develop a full 3D approach to simulate the optical properties of sharp metal tips that could be quantitatively compared with experiments and provide accurate predictions. Similar to metallic wires, elongated metal tips support a series of extended mode resonances whose properties are largely determined by the tip geometry and are essentially independent of the tip matetial. For sufficiently long tips, these resonances are energetically well separated from a broader, high-energy feature, whose position and line shape are independent of the tip length but vary strongly with the tip material. The latter is a localized plasmon mode characterized by the hemisphere terminating the tip.