We study the shapes of Milky Way satellites in the context of the tidal stirring scenario for the formation of dwarf spheroidal galaxies. The standard procedures used to measure shapes involve smoothing and binning of data and thus may not be sufficient to detect structural properties such as bars, which are usually subtle in low surface brightness systems. Taking advantage of the fact that in nearby dwarfs photometry of individual stars is available, we introduce discrete measures of shape based on the two-dimensional inertia tensor and the Fourier bar mode. We apply these measures of shape first to a variety of simulated dwarf galaxies formed via tidal stirring of disks embedded in dark matter halos and orbiting the Milky Way. In addition to strong mass loss and randomization of stellar orbits, the disks undergo morphological transformation that typically involves the formation of a triaxial bar after the first pericenter passage. These tidally induced bars persist for a few Gyr before being shortened toward a more spherical shape if the tidal force is strong enough. We test this prediction by measuring in a similar way the shape of nearby dwarf galaxies, satellites of the Milky Way. We detect inner bars in Ursa Minor, Sagittarius, Large Magellanic Cloud, and possibly Carina. In addition, 6 out of 11 dwarfs that we studied show elongated stellar distributions in the outer parts that may signify transition to tidal tails. We thus find the shapes of Milky Way satellites to be consistent with the predictions of the tidal stirring model.