Heat-capacity measurements are a useful tool for understanding the complex phase behaviour of systems containing one-dimensional motifs. Here we study the signature within such measurements of the incorporation of defects into quasi-one-dimensional (q-1D) systems. Using Monte Carlo simulations, we show that, on dilution by non-interacting sites, q-1D Ising models display a low-temperature signature not present in conventional three-dimensional models. Frustrated embeddings of 1D chains show similar features to unfrustrated embeddings, with the additional emergence of a finite ground-state entropy in the former, which arises from chain segmentation. We introduce a mean-field formulation which captures the low-temperature behaviour of the model. This theoretical framework is applied to the interpretation of experimental heat-capacity measurements of the Tb$_{1-\rho}$Y$_{\rho}$(HCOO)$_3$ family of magnetocalorics. We find good correspondence between simulation and experiment, establishing this system as a canonical example of dilute q-1D magnets.
