Cell-autonomous and self-sustained molecular oscillators drive circadian behavior and physiology in mammals. From rhythms recorded in cultured fibroblasts we identified the dominant cause for amplitude reduction as desynchronization of self-sustained oscillators. Here, we propose a general framework for quantifying luminescence signals from biochemical oscillators, both in populations and individual cells. Our model combines three essential aspects of circadian clocks: the stability of the limit cycle, fluctuations, and intercellular coupling. From population recordings we can simultaneously estimate the stiffness of individual frequencies, the period dispersion, and the interaction strength. Consistent with previous work, coupling is found to be weak and insufficient to synchronize cells. Moreover, we find that frequency fluctuations remain correlated for longer than one clock cycle, which is confirmed from individual cell recordings. Using genetic models for circadian clocks, we show that this reflects the stability properties of the underlying circadian limit-cycle oscillators, and we identify biochemical parameters that influence oscillator stability in mammals. Our study thus points to stabilizing mechanisms that dampen fluctuations to maintain accurate timing in peripheral circadian oscillators.