I have wondered for years about which of these is the correct form.

Problem:
The fifty state governments in the USA are inconsistent in the way that they refer to licenses issued to drivers.

Explanation:
I searched the Web for the form used by each of the fifty state governments.

Some state government had two official ways to refer to these licenses. Most state governments had only one way. No state government used “Drivers License” (plural, with no possessive apostrophe).

Here are the names of the states whose governments use one or more of the three forms on their own websites:

• “Driver License” (35 states):  Alabama, Alaska, Arizona, California, Colorado, Florida, Georgia, Hawaii, Illinois, Indiana (also in “Driver’s License” list), Kentucky, Massachusetts, Michigan, Mississippi, Missouri, Montana, Nebraska, Nevada, New Hampshire, New Jersey, New Mexico, New York, North Carolina, North Dakota, Ohio, Oklahoma, Oregon, Pennsylvania (also in “Driver’s License” list), South Carolina, South Dakota, Tennessee, Texas, Utah, Washington, Wisconsin
• “Driver’s License” (16 states):  Arkansas, Connecticut, Idaho, Indiana (also in “Driver License” list), Iowa, Kansas, Louisiana, Maine, Maryland, Minnesota, Pennsylvania (also in “Driver License” list), Rhode Island, Vermont, Virginia, West Virginia, Wyoming
• “Drivers’ License” (1 state):  Delaware

Ignoring the two “undecided” states (Indiana and Pennsylvania), we see that “Driver License” is favored by a ratio of more than 2.35 to 1 over “Driver’s License” (thirty-three states vs. fourteen states).

Solution:
I doubt that the fifty state governments ever will agree on a single form for referring to licenses that they issue to drivers. The best for which we can hope is that any given state government with multiple ways will settle on a single form and that Delaware will come to its senses about “Drivers’ License”.