  We consider projective rational strong Calabi dream surfaces: projective
smooth rational surfaces which admit a constant scalar curvature K\"ahler
metric for every K\"ahler class. We show that there are only two such rational
surfaces, namely the projective plane and the quadric surface. In particular,
we show that all rational surfaces other than those two admit a destabilising
slope test configuration for some polarization, as introduced by Ross and
Thomas. We further show that all Hirzebruch surfaces other than the quadric
surface and all rational surfaces with Picard rank $3$ do not admit a constant
scalar curvature K\"ahler metric in any K\"ahler class.
