Self-supervised speech models learn effective representations of spoken language, which have been shown to reflect various aspects of linguistic structure. But when does such structure emerge in model training? We study the encoding of a wide range of linguistic structures, across layers and intermediate checkpoints of six Wav2Vec2 and HuBERT models trained on spoken Dutch. We find that different levels of linguistic structure show notably distinct layerwise patterns as well as learning trajectories, which can partially be explained by differences in their degree of abstraction from the acoustic signal and the timescale at which information from the input is integrated. Moreover, we find that the level at which pre-training objectives are defined strongly affects both the layerwise organization and the learning trajectories of linguistic structures, with greater parallelism induced by higher-order prediction tasks (i.e. iteratively refined pseudo-labels).