Any couple applying for a marriage license should, in most instances, be entitled to take an affirmation. County clerks and their staff may be poorly trained and may not be familiar with your rights, but you can inform them. They may not even be familiar with what an affirmation is! When FFRF co-presidents Dan Barker and Annie Laurie Gaylor applied for their marriage license in Dane County, Wis., many years ago, they were confronted with a religious oath. When they objected and asked for a secular affirmation, the startled clerk changed the wording from "I swear" to "I affirm," then again ended the statement, "So help me God"! They had to instruct her in the definition of affirmation, but on the third try, she got it right. So stand up for your rights.
Every state may be slightly different. To find out the exact wording in your state statutes, call up your state senator or representative and ask to be given a copy of your state's laws on affirmations and alternatives to secular oaths. Many states have a legislative reference bureau or its equivalent, which can also direct you to the current statutory wording. (You can search online, but be aware that old laws can stay on the books and be misleading.)
The U.S. Constitution prohibits a religious test for public office in Article 6: "no religious test shall ever be required as a qualification to any office or public trust." Judicial decisions apply. The U.S. Supreme Court issued a unanimous ruling in Torcaso v. Watkins, 367 U.S. 448 (1961), in a case brought by a nonbeliever in Maryland who was given a religious oath when he applied to become a notary public. The Supreme Court ruled that the First Amendment to the U.S. Constitution, which is applied to state and local governments by Amendment 14, prohibits government was forcing a person to profess a belief in God or in any religion as a condition of obtaining public services or benefits. Despite the Torcaso decision, a few states are holdouts and have retained unconstitutional religious tests for public office. However, there have been a variety of lower-court decisions on the matter, including People v. Velarde, 616P.2d 104 (1980), in which the Colorado Supreme Court held that, although jurors are required to state affirmations, they are not required to swear an oath to a god. Nor should any prospective couples seeking marriage licenses!