Classes of spontaneous symmetry breaking at zero and low magnetic fields in single quantum dots (QD's) and quantum dot molecules (QDM's) are discussed in relation to the ratio R-W between the interelectron Coulomb repulsion and the harmonic confinement, using spin-and-space unrestricted Hartree-Fock calculations. These include Wigner crystallization for R-W > 1, and formation of noncrystallized electron puddles localized on the individual dots in QDM's, as well as spin-density waves in single QD's, for R-W < 1.