By simulating the filamentation of femtosecond laser pulses with flat top and different steep leading and tailing edges in air, the influence of pulse edges on the filamentation dynamics and supercontinuum (SC) generation is analyzed. It is found that the filamentation length is determined by the leading edge. The different roles of the pulse edges in the SC generation are further revealed. Under a relative lower laser power, the spectral broadening of the laser pulse originates mainly from the self-phase modulation of both pulse edges induced by the Kerr effect, while under a relative higher power condition, the leading edge plays a dominant role in the blue-side extension of the SC due to electron generation. [ABSTRACT FROM AUTHOR]