Intermediate-scale spurs are common in spiral galaxies, but perhaps most distinctively evident in a recent image showing a quasi-regular series of dust lanes projecting from the arms of M51 (Scoville & Rector 2001). We investigate, using time-dependent numerical MHD simulations, how such spurs could form (and subsequently fragment) from the interaction of a gaseous interstellar medium with a stellar spiral arm. We model the gaseous medium as a self-gravitating, magnetized, differentially-rotating, razor-thin disk. The basic flow shocks and compresses as it passes through a local segment of a tightly-wound, trailing stellar spiral arm, modeled as a rigidly-rotating gravitational potential. We first construct one-dimensional profiles for flows with spiral shocks. When the post-shock Toomre parameter $\Qsp$ is sufficiently small, self-gravity is too large for one-dimensional steady solutions to exist. The critical values of $\Qsp$ are $\sim$0.8, 0.5, and 0.4 for our models with zero, sub-equipartition, and equipartition magnetic fields, respectively. We then study the growth of self-gravitating perturbations in fully two-dimensional flows, and find that spur-like structures rapidly emerge in our magnetized models. We associate this gravitational instability with the magneto-Jeans mechanism, in which magnetic tension forces oppose the Coriolis forces that would otherwise prevent the coalescence of matter along spiral arms. The shearing and expanding velocity field shapes the condensed material into spurs as it flows downstream from the arms. Although we find swing amplification can help form spurs when the arm-interarm contrast is moderate, unmagnetized systems that are quasi-axisymmetrically stable are generally also stable to nonaxisymmetric perturbations, suggesting that magnetic effects are essential. In nonlinear stages of evolution, the spurs in our models undergo fragmentation to form $\sim4\times 10^6 \Msun$ clumps, which we suggest could evolve into bright arm and interarm \ion{H}{2} regions as seen in spiral galaxies.

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