Fractal processes, like trees or coastlines, are defined by self-similarity or power law scaling controlled by a single exponent, simply related to the fractal dimension or Hurst exponent (H) of the process. Multifractal processes, like turbulence, have more complex behaviours defined by a spectrum of possible local scaling behaviours or singularity exponents (h). Here, we report two experiments that explore the relationships between instrumental and cognitive variables and the monofractal and multifractal parameters of functional magnetic resonance imaging (fMRI) data acquired in a no-task or resting state. First, we show that the Hurst exponent is greater in grey matter than in white matter regions, and it is maximal in grey matter when data were acquired with an echo time known to optimise BOLD contrast. Second, we show that latency of response in a fame decision/facial encoding task was negatively correlated with the Hurst exponent of resting state data acquired 30 min after task performance. This association was localised to a right inferior frontal cortical region activated by the fame decision task and indicated that people with shorter response latency had more persistent dynamics (higher values of H). Multifractal analysis revealed that faster responding participants had wider singularity spectra of resting fMRI time series in inferior frontal cortex. Endogenous brain oscillations measured by fMRI have monofractal and multifractal properties that can be related to instrumental and cognitive factors in a way, which indicates that these low frequency dynamics are relevant to neurocognitive function.