Is bachelorette replacing spinster and other synonyms?

Solution 1:

According to COCA, bachelorette is more popular than spinster (and unmarried woman), something that only really changed in the last 10-20 years. Compare the "per mil" rows in the following charts.

Spinster:

Bachelorette:

Unmarried Woman:

One pretty significant thing I see in these charts is the fact that most of the hits for spinster are from fiction (and there are very few hits for the other sections).

It's also important to note that spinster is not neutral:

The development of the word spinster is a good example of the way in which a word acquires strong connotations to the extent that it can no longer be used in a neutral sense. From the 17th century the word was appended to names as the official legal description of an unmarried woman: Elizabeth Harris of London, Spinster. This type of use survives today in some legal and religious contexts. In modern everyday English, however, spinster cannot be used to mean simply ‘unmarried woman’; it is now always a derogatory term, referring or alluding to a stereotype of an older woman who is unmarried, childless, prissy, and repressed
_{Oxford Dictionaries}

Solution 2:

Is bachelorette becoming the "preferred" neutral expression to refer to an "unmarried woman"?

These ngrams, first AmE and second BrE do not suggest it is so:

Is it commonly used both in BrE and AmE? & what may have caused its increased in usage from the '90s, one century later than it was first coined?

In the U.S. the eponymously named show has likely had influence. Spinster may have declined somewhat and likely suffers from the pc police. Single woman seems to be more acceptable. In the U.S. rarely is bachelorette used unless discussing the TV show or in just casual/informal talk.

In this wikipedia article the possible origins of bachelorette are discussed (1965 and 2003). wikipedia

and this article on the BrE use of same: Anglophenia

addendum: I coaxed ngram to 'cough up' data to 2008 v 2000. (AmE on the left, BrE on the right) There is noted a definite uptick in bachelorette but single woman still predominates. But 10 years have passed. There is a sophisticated too to analyze spoken word, (BYU Corpora) but it is costly.