The authorities said they did not know of any arrests yet from frauds related to the new health care law. But they said cases were being investigated by the police and the antifraud divisions of state insurance departments.

The Nevada Independent Insurance Agents, a trade organization, has collected 25 different fax messages offering bogus health coverage, sometimes for as little as $29.95 per month.

“A $10 or $40 prescription card included,” advertises one fax, which contains the image of a caduceus, the emblem for doctors, but no company name.

“We can also provide dental care for you,” says another that bears an image of an ugly tooth and offers customer service in Spanish.

Kay Lockhart, the organization’s president, said the term Obamacare was popular among dishonest marketers because it was familiar to the public; the Tea Party movement uses the term to refer derisively to the health care law.

In response to the frauds, the Nevada organization has started running advertisements statewide that spoof an imaginary fraudulent company called Swindol, Frod and Rhipoff Health Insurance.

“Not far from the truth,” the advertisements warn. “Question the company and their plan.”

In Mobile, Ala., Tina Waller, the president of the Better Business Bureau of South Alabama, said she received questions earlier this month from a middle-aged woman who was offered “one of the last 20 spots” on a government health insurance plan. The woman did not have insurance and had learned about the discount plan from a friend who bought it from a telemarketer. Ms. Waller investigated the company, concluded it was fraudulent and reported it to the authorities.