Quantum-inspired methods via tensor network algorithms carry a two-sided expectation into the field of computational fluid dynamics (CFD): alternative algorithms for classical computing hardware that leverage the underlying compression scheme of tensor networks, and a pool of approaches that enable and drive the development of quantum algorithms for quantum processing units (QPUs). To close the gap to current state-of-the-art CFD algorithms, we present tensor network algorithms for compressible flows guided by the Navier-Stokes and Euler equations. We present our approach for integrating the compressible Navier-Stokes equations using matrix product states (MPS) and the MacCormack time-stepping scheme. The test case considered is the 2D periodic shear flow for different physical parameters. We then introduce simplifications leading to the compressible Euler equations and discuss paradigmatic flow examples in one and two spatial dimensions with non-periodic boundary conditions. These test cases include the Rayleigh-Taylor instability and other shock-free or shock-developing cases; they allow us to pinpoint the regime in which this approach works successfully. Our conclusion contains a detailed discussion of how the bond dimension influences computational performance across different steps and scales with the flow parameters. We identify the computational bottlenecks and discuss how they can be overcome, e.g., by using graphics processing units (GPUs) as accelerators. Further, we outline the hardware requirements for running corresponding quantum algorithms on quantum hardware and comment on possible performance gains.