In a stationary axisymmetric spacetime, the angular velocity of a stationary observer whose acceleration vector is Fermi-Walker transported is also the angular velocity that locally extremizes the magnitude of the acceleration of such an observer. The converse is also true if the spacetime is symmetric under reversing both t and together. Thus a congruence of non-rotating acceleration worldlines (NAW) is equivalent to a stationary congruence accelerating locally extremely (SCALE). These congruences are defined completely locally, unlike the case of zero angular momentum observers (ZAMOs), which requires knowledge around a symmetry axis. The SCALE subcase of a stationary congruence accelerating maximally (SCAM) is made up of stationary worldlines that may be considered to be locally most nearly at rest in a stationary axisymmetric gravitational field. Formulae for the angular velocity and other properties of the SCALEs are given explicitly on a generalization of an equatorial plane, infinitesimally near a symmetry axis, and in a slowly rotating gravitational field, including the far-field limit, where the SCAM is shown to be counter-rotating relative to infinity. These formulae are evaluated in particular detail for the Kerr-Newman metric. Various other congruences are also defined, such as a stationary congruence rotating at minimum (SCRAM), and stationary worldlines accelerating radially maximally (SWARM), both of which coincide with a SCAM on an equatorial plane of reflection symmetry. Applications are also made to the gravitational fields of maximally rotating stars, the Sun and the Solar System.