We investigate the Heisenberg model on a decorated square (Fisher) lattice in the presence of first-neighbor J1, second-neighbor J2, and third-neighbor J3 exchange couplings, with antiferromagnetic J1. The classical ground-state phase diagram obtained within a Luttinger-Tisza framework is spanned by two antiferromagnetically ordered phases, and an infinitely degenerate antiferromagnetic chain phase. Employing classical Monte Carlo simulations we show that thermal fluctuations fail to lift the degeneracy of the antiferromagnetic chain phase. Interestingly, the spin-wave spectrum of the Néel state displays three Dirac nodal loops out of which two are symmetry protected while for the antiferromagnetic chain phase we find symmetry-protected Dirac lines. Furthermore, we investigate the spin S=12 limit employing a bond operator formalism which captures the singlet-triplet dynamics, and find a rich ground-state phase diagram host to a variety of valence bond solid orders in addition to antiferromagnetically ordered phases. © 2020 American Physical Society.