In this paper, we report on results from a computational investigation of nanosecond pulsed surface discharges of positive and negative polarity using a 2D fluid and fluid-Monte Carlo simulation. The streamers propagate along the dielectric surface in an asymmetric actuator geometry. The essential difference between the streamers of positive and negative polarities is observed. For positive polarity the intense sheath region is formed near the surface having high values of the electric field. The negative streamer has a lower field at the streamer front and in the sheath region. The disparity between the positive and negative surface streamers increases when electron Monte Carlo simulation is used that treats the energetic secondary electrons in a fully kinetic way. We also found that for a negative polarity applied to the exposed electrode, a thin layer of precursor electrons ahead of a streamer is formed having a shape of a narrow protruding needle. The effect is attributed to the production and trapping of fast secondary electrons in the narrow anodic sheath region.