One of the main goals of archeomagnetism is to document the secular changes of Earth's magnetic field by laboratory analysis of the magnetization carried by archeological artefacts. Typical techniques for creating a time-dependent model assume a prescribed temporal discretization which, when coupled with sparse data coverage, require strong regularization generally applied over the entire time-series in order to ensure smoothness. Such techniques make it difficult to characterize uncertainty and frequency content, and robustly detect rapid changes. Key to proper modelling (and physical understanding) is a method that places a minimum level of regularization on any fit to the data. Here we apply a transdimensional Bayesian technique based on piecewise linear interpolation to sparse archeointensity data sets, in which the temporal complexity of the model is not set a priori, but is self-selected by the data. The method produces two key outputs: (i) a posterior distribution of intensity as a function of time, a useful tool for archeomagnetic dating, whose statistics are smooth but formally unregularized and (ii) by including the data ages in the model of unknown parameters, the method also produces posterior age statistics of each individual contributing datum. We test the technique using synthetic data sets and confirm agreement of our method with an integrated likelihood approach. We then apply the method to three archeomagnetic data sets all reduced to a single location: one temporally well-sampled within 700 km from Paris (here referred to as Paris700), one that is temporally sparse centred on Hawaii, and a third (from Lübeck, Germany and Paris700) that has additional ordering constraints on age from stratification. Compared with other methods, our average posterior distributions largely agree, however our credible intervals appear to much better reflect the uncertainty during periods of sparse data coverage. Because each ensemble member of the posterior distribution is piecewise linear, we only fit oscillations when required by the data. As an example, we show that an oscillatory signal, associated with temporally localized intensity maxima reported for a sparse Hawaiian data set, is not required by the data. However, we do recover the previously reported oscillation of period 260 yr for the Paris700 data set and compute the probability distribution of the period of oscillation. We further demonstrate that such an oscillation is unresolved when accounting for age uncertainty by using a fixed age and with an artificially inflated error budget on intensity.