For a two-dimensional electron gas (2DEG) with a nonabrupt edge electron-density profile, edge modes, analogous to the higher-multipole surface modes of a 3DEG, are found in addition to the regular edge plasmon. Several simple model edge density profiles are investigated. For a linear edge profile the integral equation for the scalar potential is solved by expanding in a complete orthonormal set of functions. The resulting secular equation is truncated and solved numerically. For a simple double-step edge profile, which can be solved analytically, the multipole edge modes can be understood in a simple intuitive way.