When no training or adaptation data is available, semi-supervised training is a good alternative for processing new domains. We perform Bayesian training of a part-of-speech (POS) tagger from unannotated text and a dictionary of possible tags for each word. We extend that method with supervised prediction of possible tags for out-of-vocabulary words and study the impact of both semi-supervision and starting dictionary size on three representative downstream tasks (named entity tagging, semantic role labeling, ASR output post-processing) that use POS tags as features. The outcome is no impact or a small decrease in performance compared to using a fully supervised tagger, with even potential gains in case of domain mismatch for the supervised tagger. Tasks that trust the tags completely (like ASR post-processing) are more affected by a reduction of the starting dictionnary, but still yield positive outcome.