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Silicon carbide has recently emerged as a promising photonics material due to its unique properties, including possessing strong second- and third-order nonlinear coefficients and hosting various color centers that can be utilized for a wealth of quantum applications. Here, we report the design and demonstration of octave-spanning microcombs in a 4H-silicon-carbide-on-insulator platform for the first time, to our knowledge. Such broadband operation is enabled by optimized nanofabrication achieving$>1$million intrinsic quality factors in a 36-μm-radius microring resonator, and careful dispersion engineering by investigating the dispersion properties of different mode families. For example, for the fundamental transverse-electric mode whose dispersion can be tailored by simply varying the microring waveguide width, we realized a microcomb spectrum covering the wavelength range from 1100 nm to 2400 nm with an on-chip power near 120 mW. While the observed comb state is verified to be chaotic and not soliton, attaining such a large bandwidth is a crucial step towards realizing$f-2f$self-referencing. In addition, we also observed a coherent soliton-crystal state for the fundamental transverse-magnetic mode, which exhibits stronger dispersion than the fundamental transverse-electric mode and hence a narrower bandwidth.