We present a survey of the structural and magnetic properties of submonolayer transition metal dioxides on the (001) surfaces of the heavy face-centered cubic (fcc) noble metals Ir and Pt performed by spin-averaged scanning tunneling microscopy (STM) and spin-polarized (SP-)STM. Our STM results confirm that deposition of Co, Fe, Mn, and Cr on the (2 × 1) oxygen-reconstructed Ir(001) surface leads to the formation of quasi one-dimensional chains with a (3 × 1) unit cell. As recently predicted by density functional theory [Ferstl et al., Phys. Rev. Lett. 117, 046101 (2016)], our SP-STM images of FeO2 and MnO2 on Ir(001) show a two-fold periodicity along the chains which is characteristic for an antiferromagnetic coupling along the chains. In addition, these two materials also exhibit spontaneous, permanent, and long-range magnetic coupling across the chains. Whereas we find a ferromagnetic inter-chain coupling for FeO2/Ir(001), the magnetic coupling of MnO2 on Ir(001) appears to be a non-collinear 120 • spin spiral, resulting in a (9 × 2) magnetic unit cell. On Pt(001) patches of (3 × 1)-reconstructed oxide chains could only be prepared by transition metal (Co, Fe, and Mn) deposition onto the cold substrate and subsequent annealing in an oxygen atmosphere. Again SP-STM on MnO2/Pt(001) reveals a very large, (15 × 2) magnetic unit cell which can tentatively be explained by a commensurate 72 • spin spiral. Large scale SP-STM images reveal a long wavelength spin rotation along the MnO2 chain.