Single cell RNA sequencing (scRNASeq) data plays a major role in advancing our understanding of developmental biology. An important current question is how to classify transcriptomic profiles obtained from scRNASeq experiments into the various cell types and identify the lineage relationship for individual cells. Because of the fast accumulation of datasets and the high dimensionality of the data, it has become challenging to explore and annotate single cell transcriptomic profiles by hand. To overcome this challenge, automated classification methods are needed. Classical approaches rely on supervised training datasets. However, due to the difficulty of obtaining data annotated at single cell resolution, we propose instead to take advantage of partial annotations. The partial label learning framework assumes that we can obtain a set of candidate labels containing the correct one for each data point, a simpler setting than requiring a fully supervised training dataset. We study and extend when needed state of the art multi-class classification methods, such as SVM, k-NN and prototype-based methods, to the partial label learning framework. Moreover, we study the effect of incorporating the structure of the label set into the methods. We focus particularly on the hierarchical structure of the labels, as commonly observed in developmental processes. We show, on simulated and real datasets, that these extensions enable to learn from partially labeled data, and perform predictions with high accuracy, particularly with a non-linear prototype-based method. We demonstrate that the performances of our methods trained with partially annotated data reaches the same performance than fully supervised data. Finally, we study the level of uncertainty present in the partially annotated data, and derive some prescriptive results on the effect of this uncertainty on the accuracy of the partial label learning methods. Overall our findings show how hierarchical and non-hierarchical partial label learning strategies can help solve the problem of automated classification of single cell transcriptomic profiles, interestingly these methods rely on a much less stringent type of annotated datasets compared to fully supervised learning methods.