Axisymmetric equilibrium shapes and stability of linearly polarizable dielectric (ferrofluid) drops of fixed volume which are pendant/sessile on one plate of a parallel-plate capacitor and are subjected to an applied electric (magnetic) field are determined by solving simultaneously the free boundary problem comprised of the Young-Laplace equation for drop shape and the Laplace equation for electric (magnetic) field distribution. When the contact angle that the drop makes with the plate is fixed to be 90° and the distance between the plates is infinite, the results are identical to those of a free drop immersed in a uniform field and resolve discrepancies between previously reported theoretical predictions and experimental measurements. Remarkably, regardless of the value of the ratio of the permittivity (permeability) of the drop to that of the surrounding fluid, κ, drop shapes develop conical tips as drop deformation increases. However, three types of behaviour are found, depending on the value of κ. When κ < κ1, the drop deformation grows without bound as field strength rises. On the other hand, when κ > κ2 > κ1, families of equilibrium drop shapes become unstable at turning points with respect to field strength. Beyond the turning points, the unstable families terminate: the mean curvature at the virtually conical drop tip grows without bound. However, in the range κ1 < κ < κ2, the new results predict that drop deformation exhibits hysteresis, in accord with experiments of Bacri, Salin & Massart (1982) and Bacri & Salin (1982, 1983). Such hysteresis phenomena have been surmized previously on the basis of approximate theories, though they have not been calculated systematically until now. Moreover, detailed computations reveal the importance of varying the drop size and plate spacing, and whether, along the three-phase contact line, the contact line is fixed or the contact angle is prescribed.