In nature, variations in mantle sources and magmatic processes lead to significant changes of crustal thickness distribution and spreading pattern at mid-ocean ridges. Distributions in crust thicknesses are relatively uniform at fast ridges. On the contrary, great lateral fluctuations with even mantle rocks exhumation are observed at slow and ultraslow ridges. Similarity, the spreading pattern changes from symmetric configuration (at fast ridges) to highly asymmetric plates accretion with detachment faults and oceanic core complexes (at slow ridges). Recent modeling studies suggested that magmatism may play a key role in the variation. Yet, the physical mechanisms controlling spatial-temporal distribution and intensity of the magmatic activity at mid-ocean ridges remain elusive. In this study, using 3D self-consistent magmatic-thermomechanical numerical models, we systematically investigate the effect of mantle potential temperature and spreading rate on the crustal thickness lateral variations and tectonic pattern at spreading ridges of various opening rates. Our numerical results show that along melt-poor slow-ultraslow mid-ocean ridges, there are two fundamentally different types of ridge sections that spontaneously form and alternate. This is due to a tectono-magmatic instability which self-consistently partitions melt supply beneath the ridge. Two geometries form: (1) normal ridge sections (NR) with elevated topography, normal thickness of oceanic crust and a hot thermal structure, and (2) fracture zone sections (FZ) with lowered topography, thin/absent crust, exhumed mantle rocks, and a cold thermal structure. The tectono-magmatic instability along spreading ridges is triggered by the combined effects of a reduced melt supply from the mantle into crustal magma chambers and an increase in brittle layer thickness. Such bimodal structure is then maintained by the combined effects of buoyant melt partitioning along the ridge, localized latent heat release from crustal magma crystallization, and different spreading modes. This results, on one hand, in hotter, thinner and elevated magmatic sections (tectono-magmatic spreading), and, on the other hand, in colder, thicker and subsided amagmatic sections (purely tectonic spreading). This predicted bimodal distribution finds a good agreement with natural observations. This suggests that spontaneous self-organization of magma supply controlled by spreading rate and mantle potential temperature plays a critical role in shaping tectonic and crustal patterns of mid-ocean ridges.