Larger shares of electricity generation based on volatile renewables often lead to high curtailment rates and thus a loss of carbon-neutral energy. Small-scale residential power-to-heat applications can help to improve this situation by flexibly increasing electricity demand and thus integrating otherwise curtailed renewable power production. Nevertheless, to include such flexibilities in overall system operation there is a need for a reliable quantitative planning and action basis. For that, we propose a method to perform probabilistic day-ahead forecasts of available thermal storage capacities for residential power-to-heat operation based on artificial neural networks. The prediction is structured as two-step approach, consisting of a day-ahead prediction of storage temperatures and a subsequent derivation of available storage capacities. In order to better address uncertainties in the residential sector, probabilistic forecasts are carried out. For the temperature prediction a neural network structure consisting of long short-term memory layers and a mixture density output is used. The predicted probability distributions of storage temperatures are subsequently sampled and transformed to probability distributions of storage capacities. To ensure suitable hyper-parameter configurations, an automated optimization of these parameters is carried out. For a demonstration of the general applicability of the approach a case study is performed based on data of a single-family household in northern Germany. We compare the approach to different deterministic and probabilistic benchmark forecasting models, showing that the proposed approach clearly outperforms the benchmark models.