The United States Constitution as interpreted by the U.S. Supreme Court is clear. You may not incarcerate someone for inability to pay a sum of money, civil or criminal, unless they are able to do so, but are willfully refusing to do so (e.g. if the debtor has control over a foreign bank account over which the court cannot obtain jurisdiction).
In practice, people who don't pay fine only offenses (e.g. parking and traffic tickets and minor hunting, fishing and ordinance violations) are routinely jailed for failure to pay fines. The dollar amounts look tiny to members of the middle class, but can be insurmountable to those who are on the margins (which makes up a disproportionate share of these debtors).
A testimonial from a Texas municipal judge explains why this is unconstitutional and what judges should be doing instead.
Another circumstance in which incarceration for unpaid debts is common involves remedial contempt citations entered against self-employed parents who owe child support and don't pay as agreed. In some unusual circumstances, this is proper (and some states have criminal laws calling for incarceration sentences for essentially identical conduct). But, this also produces little compliance, because usually, by the time a hearing comes around the self-employed person has spent the money and can't deliver it and really shouldn't as a result, be held in remedial contempt incarceration at all.
A third circumstance in which people are routinely incarcerated basically for being poor, is when someone is arrested for a minor offense, bail is set automatically or as a matter of course by the judge in some small amount without regard to the ability of the defendant to pay, and the defendant doesn't have the resources to make bail on the often trivially minor charge. Typically, something like 90% of indigent defendants can be released on personal recognizance bonds without any worse outcomes than the traditional bail system.
Of course, any incarceration, can led to lots of very bad outcomes: loss of a job or for the self-employed loss of business, loss of custody of children or the turnover of children to foster care, coerced pleads to criminal charges for which defendants are not guilty, homelessness upon release (often with loss of personal property if it is lost in an eviction as is common), interruptions in medical care, invasive strip searches, injuries at the hands of guards or fellow inmates who are incarcerated for more serious and often violent offenses, suicide, and high governments costs for incarceration and the related legal process.
Any just system of justice needs to do a much better job, systemically, of handling these kinds of issues that senselessly and harmfully increase the incarceration rate where there is no sound legal or policy argument for doing so. These cases are the low hanging fruit of sentencing reform.