Characterizing the variability across timescales is important for understanding the underlying dynamics of the Earth system. It remains challenging to do so from palaeoclimate archives since they are more often than not irregular, and traditional methods for producing timescale-dependent estimates of variability, such as the classical periodogram and the multitaper spectrum, generally require regular time sampling. We have compared those traditional methods using interpolation with interpolation-free methods, namely the Lomb-Scargle periodogram and the first-order Haar structure function. The ability of those methods to produce timescale-dependent estimates of variability when applied to irregular data was evaluated in a comparative framework, using surrogate palaeo-proxy data generated with realistic sampling. The metric we chose to compare them is the scaling exponent, i.e. the linear slope in log-transformed coordinates, since it summarizes the behaviour of the variability across timescales. We found that, for scaling estimates in irregular time series, the interpolation-free methods are to be preferred over the methods requiring interpolation as they allow for the utilization of the information from shorter timescales which are particularly affected by the irregularity. In addition, our results suggest that the Haar structure function is the safer choice of interpolation-free method since the Lomb-Scargle periodogram is unreliable when the underlying process generating the time series is not stationary. Given that we cannot know a priori what kind of scaling behaviour is contained in a palaeoclimate time series, and that it is also possible that this changes as a function of timescale, it is a desirable characteristic for the method to handle both stationary and non-stationary cases alike.