Many of the models of dark galactic halos described in previous sections assume that the density profile can be described by some kind of isothermal sphere. As discussed in Section 6, steeper density profiles are also consistent with much of the data, but there is the more radical possibility that halos are significantly flattened. Some degree of flattening would not be a great surprise, since stellar systems exhibit varying degrees of oblateness or triaxiality. However, the difficulties in pinning down the mass distribution of halos discussed in earlier sections suggests that establishing the shape of halos is an even bigger problem. Despite the magnitude of the task, some progress has been made.
An early argument for the existence of dark halos, and spherical halos in particular, was given by Ostriker and Peebles (1973). They suggested that dark spherical mass distributions around spirals would supress bar instabilities in their stellar disks. This argument was undermined by Kalnajs (1983), who showed that a central bulge or hot disk could do the same job just as effectively (see also Sellwood 1985). Monet, Richstone and Schechter (1981) argued that the roundness of the Galactic bulge implied the presence of a roughly spherical dark halo around the Milky Way. Hartwick (1987) reached a similar conclusion from the spatial distribution of RR Lyraes and metal-poor globular clusters in the Galactic halo. Dupraz and Combes (1986) proposed that the shapes of shells observed around some galaxies could be used to investigate the halo flattening. However, N-body simulations that reproduce such shells in galaxy mergers show that the shell shape probably does not reflect the halo shape (Hernquist and Quinn 1988). Twisted velocity fields in the HI disks of spirals may also reflect a non-spherical halo, although to definitively determiine the halo shape it is necessary to decouple kinematic twists from possible structural twists. Rotation curve modelling in spirals can provide some shape information (Carignan DMW), but high-quality, extended HI curves are needed. The shape of the bulge in spiral galaxies may reflect competition between the flattening effects of the disk potential and the rounder halo, but the bulge shape is also sensitive to rotation and anisotropic velocity dispersions.