Those hypersurfaces admitting a 1-parameter symmetry group are characterised by failure of versality of a certain unfolding of their set of singularities, which in the simplest cases (sextic curves, quartic surfaces and cubic 4-folds) is the unfolding by hypersurfaces of the same degree. We give a classification of these hypersurfaces, and calculate their total Milnor and Tjurina numbers. The maximal Tjurina number occurs if and only if the equation (of degree d) is annihilated by a vector field of degree (d-2) independent of that given by the action; in these cases the enumeration is more explicit, and when also d=3 we have a 2-parameter group. It is conjectured, and proved in low dimensions, that any hypersurface with maximal Tjurina number admits a 1-parameter symmetry group.