The scattering of ultrashort and ultraintensie laser pulses by single electrons has been investigated theoretically and numerically by use of the classical theory of Thomson scattering by free electrons. The results indicate that attosecond pulse trains are emitted during the interaction. The temporal and spatial characteristics of the radiation are presented for different laser parameters(including intensity, pulse duration, initial phases and polarizations) and initial states of the electron(including different initial velocities and positions). Usually, with the increase of the incident laser intensity, the radiation becomes more powerful, its central frequency becomes higher, its duration becomes shorter, and its angular distribution becomes smaller. The radiation produced by the electron under a linearly polarized laser has a higher amplitude than under a circularly polarized laser pulse with the same intensity. In both cases, the polarization state of the radiation is very complicated, which depends n the observation direction. In spite of its initial energy and moving direction, the radiation produced by relativistic energetic electrons tends to be predominantly along the moving direction of the electron.