I have recently somewhat tangently answered a question that has a common point to this one, and might be useful. So any follow ups are welcome, but I will build upon a point in there where the parties colluded against the court to obtain a divorce. All the qualifications from the other answer still remain, namely, state laws were different, and changed with times, specially the application.

"It would be aiming a deadly blow at public morals to decree a dissolution of the marriage contract merely because the parties requested it," Chancellor Walworth in 1828.

Courts were aware of fraud and possibility of collusion, so generally, confessions of the parties were not deemed reliable grounds of divorce, which was generally issued only by such a severe and repugnant violation of legislative and public norms that would be damaging to the innocent party/spouse involved. And further, courts were aware that the parties sometimes commited various acts, like the sin of adultery, in collusion to meet the technically required grounds,^(1) and for divorces on such grounds, a reliable third witness was needed.

Two options;

(1) Collusion against the courts on various grounds techincally viable for divorce.
(2) Collusion, in a sense of relocation. The documentation of bigamy is substantial, and bigamy after the collusion between the original parties, insofar as this was in mutual agreement, was quite safe from being discovered. On the other hand, bigamy that resulted from disertion from one of the spouses from the original marriage, which was not agreed upon, after discovery, resulted in some rather notable trials.

Bottom line, with slight means and some wits, determined parties could get a divorce, so there were mostly other actors usually hindering such actions, like kins, relatives, broader social setting and acquaintances, et all.

I am not familiar with any accurate estimation of rate of such divorces, if it is indeed even possible to approximate one. Although all these characteristics depended on state enormously.

1 James Kent, Commentraries on American Law, 1926-30, Chief Justice of Supreme Court of New York.