Assuming you are talking about the US...

Generally, women started doing this in the mid-19th century as the rise of cities and the industrial age brought on more job opportunities that existed outside of a family homestead. Women who worked outside of the home in the early 1800s would typically board with another family, and while this was still commonplace at the end of the century, more and more women lived in boarding houses, with roommates, or even on their own. Women were paid less than their male counterparts under the false reasoning that they were always economically supported by men, and earned only for supplemental income (known as pin money). This actually increased the likelihood that women would rely on one another for economic sustenance, however. Among educated women from middle and upper classes, the term "Boston Marriage" entered the lexicon in the late 19th century to describe this type of relationship between female roommates (at least some of whom were also lovers).

With all that being said, many people still considered it taboo for a woman to live in any arrangement outside of the familial structure in the mid 20th century. For instance, universities typically allowed male students to live off campus and with roommates but kept close watch on female students and required that they board on campus or with family (often called in loco parentis policies). Rebellion against this policy, and the notion that women could not be trusted live outside the familial structure, was certainly part of the women's liberation movement of the 1960s and '70s, and by the 1980s it was entirely common for young and single American women to live with roommates.

So, there isn't really a single answer to your question-- it was OK for women to be single and have roommates according to a significant number of people for a century before it was OK in the minds of the vast majority.