The behaviour of weak laminar plane fountains that result from the injection of dense fluids upwards into a large container of homogeneous fluid of lower density is studied numerically by using a time-accurate finite volume code. The study is an extension of a previous numerical study on weak laminar axisymmetric fountains. Weak laminar plane fountains with both a uniform and a parabolic profile of the discharge velocity at the source have been investigated. The initial, temporary and final characteristic fountain heights and the times for the fountain front to reach these heights have been determined and scaled. At steady state, a height scale as well as a horizontal length scale for the fountain width are determined. The vertical distribution of the vertical velocity and temperature on the symmetry line are shown to scale with this height. The height and horizontal length scales have been used to scale the horizontal variation of both the horizontal and the vertical velocity in the fountain core at steady state.