We introduce a multilabel probing task to assess the morphosyntactic representations of word embeddings from multilingual language models. We demonstrate this task with multilingual BERT (Devlin et al., 2018), training probes for seven typologically diverse languages of varying morphological complexity: Afrikaans, Croatian, Finnish, Hebrew, Korean, Spanish, and Turkish. Through this simple but robust paradigm, we show that multilingual BERT renders many morphosyntactic features easily and simultaneously extractable (e.g., gender, grammatical case, pronominal type). We further evaluate the probes on six "held-out" languages in a zero-shot transfer setting: Arabic, Chinese, Marathi, Slovenian, Tagalog, and Yoruba. This style of probing has the added benefit of revealing the linguistic properties that language models recognize as being shared across languages. For instance, the probes performed well on recognizing nouns in the held-out languages, suggesting that multilingual BERT has a conception of noun-hood that transcends individual languages; yet, the same was not true of adjectives.