On what grounds can I apply for a hardship exemption to the individual mandate?

For the 2018 tax year, you can claim a hardship exemption if you experienced any circumstances that prevented you from obtaining coverage.  For the 2018 tax year only, you can claim the hardship exemption directly on your tax return by checking the box on Form 1040.  You will not be required to submit documentation of the hardship with your return, but you should keep any documentation for your own records.

If you are late-filing or updating a tax return for earlier years and require a hardship exemption for that year, you can apply for a hardship exemption if you experienced difficult financial or domestic circumstances that prevent you from obtaining coverage – such as homelessness, death of a close family member, bankruptcy, substantial recent medical debt, or disasters that substantially damaged your property. In addition, a hardship exemption may be granted if you were determined ineligible for Medicaid only because your state hasn’t expanded Medicaid coverage to residents with income up to 138% of the federal poverty level.  You can also apply for a hardship exemption if obtaining coverage would have been so burdensome as to cause you to experience other serious deprivation of food, shelter, or other necessities.

For tax years before 2018, you can apply to the Marketplace for a hardship exemption.  If granted, the Marketplace will give you an exemption certificate.  You should enter that certificate number in Form 8965, which you should file with the tax return.