A Jewish woman has the halachic status of niddah from menarche until immersion prior to marriage. However, as the only consequence at present of being niddah is that physical contact with men is forbidden, and halacha does not approve of physical contact with men outside of the context of marriage, this does not present halachic problems.
Unclean is an unfortunate translation of tamei. Tumah is a ritual status, not a function of bodily cleanliness. At the time that there was a Temple in Jerusalem, unmarried women did use the mikveh to allow them to participate in certain rituals. Today, in the absence of the Temple, they do not.