Laser cavity-solitons can appear in systems comprised of a nonlinear microcavity nested within an amplifying fiber loop. These states are robust and self-emergent and constitute an attractive class of solitons that are highly suitable for microcomb generation. Here, we present a detailed study of the free-running stability properties of the carrier frequency and repetition rate of single solitons, which are the most suitable states for developing robust ultrafast and high repetition rate comb sources. We achieve free-running fractional stability on both optical carrier and repetition rate (i.e., 48.9 GHz) frequencies on the order of [Formula: see text] for a 1 s gate time. The repetition rate results compare well with the performance of state-of-the-art (externally driven) microcomb sources, and the carrier frequency stability is in the range of performance typical of modern free-running fiber lasers. Finally, we show that these quantities can be controlled by modulating the laser pump current and the cavity length, providing a path for active locking and long-term stabilization.