Spinster isn't for old women, but it certainly would be more often used for that, and has a connotation of having failed to get a husband, rather than being happily unmarried.
Bachelorette is sometimes used solely for "unmarried" but does indeed sometimes include divorcées and widows. It's rarely used (was popular from about the 1930s through to the 1960s) and often considered patronising.
"Bachelor girl" is older than bachelorette, and generally only for never-married. It is now rare.
There isn't an exact equivalent, because the English language is influenced by thousands of years in which the status of different sexes both within and outside of marriage was not one of parity.