Context. Recent studies have revealed a hitherto unknown complexity of Cepheid pulsations by discovering irregular modulated variability using photometry, radial velocities, and interferometry. Aims. We aim to perform a statistically rigorous search and characterization of such phenomena in continuous time, applying it to 53 classical Cepheids from the OGLE-III catalog. Methods. We have used local kernel regression to search for both period and amplitude modulations simultaneously in continuous time and to investigate their detectability. We determined confidence intervals using parametric and non-parametric bootstrap sampling to estimate significance, and investigated multi-periodicity using a modified pre-whitening approach that relies on time-dependent light curve parameters. Results. We find a wide variety of period and amplitude modulations and confirm that first overtone pulsators are less stable than fundamental mode Cepheids. Significant temporal variations in period are more frequently detected than those in amplitude. We find a range of modulation intensities, suggesting that both amplitude and period modulations are ubiquitous among Cepheids. Over the 12-year baseline offered by OGLE-III, we find that period changes are often nonlinear, sometimes cyclic, suggesting physical origins beyond secular evolution. Our method detects modulations (period and amplitude) more efficiently than conventional methods that are reliant on certain features in the Fourier spectrum, and pre-whitens time series more accurately than using constant light curve parameters, removing spurious secondary peaks effectively. Conclusions. Period and amplitude modulations appear to be ubiquitous among Cepheids. Current detectability is limited by observational cadence and photometric precision: detection of amplitude modulation below 3 mmag requires space-based facilities. Recent and ongoing space missions (K2, BRITE, MOST, CoRoT) as well as upcoming ones (TESS, PLATO) will significantly improve detectability of fast modulations, such as cycle-to-cycle variations, by providing high-cadence high-precision photometry. High-quality long-term ground-based photometric time series will remain crucial to study longer-term modulations and to disentangle random fluctuations from secular evolution.