State laws differ however even if laws do not require your spouse to be listed on your car insurance policy your insurer may require you to include him or her on the policy.
Most insurance companies require policyholders to list all licensed household members on their policy since that these people will normally have access to your vehicles and will probably drive the cars on the policy on a frequent basis. In some states if you do not tell your insurance company of all of the drivers in your household it can be deemed misrepresentation, a form of fraud. The state laws extend liability to defined household member so insurance companies have to ask and extend coverage to them too.
If you do not want your spouse on your policy then you may be able to have him or her excluded from the policy, if state laws and your insurance company's guidelines allow. Keep in mind though if you exclude your spouse and he or she does drive your car and gets into an accident your policy's coverage will not be extended to him or her in anyway.
In general, insurance companies do not tend to want to exclude a spouse who has been licensed so normally you will need to have your spouse on your policy. You can get a quote with Safeco, Progressive, or Direct General on our site though and when purchasing, select the option to not include that driver on the policy and select the appropriate exclusion options to get an accurate rate. If your state and their underwriting guidelines for your situation allows you should be able to exclude your spouse.
Remember that if you did exclude your spouse then none of the coverages under your car insurance would extend to him or her in any way if he/she was to drive the vehicle(s) listed on your policy and cause an accident. So even in an emergency situation he/she should not drive your car if you exclude your spouse from your policy.
If you do not want to exclude your spouse then he/she will need to be on the policy normally. If your spouse has a car of their own to insure and it is on a different policy then putting both your vehicles on the same auto insurance policy could save you money with a multi-car policy instead of each having their own policy. You and your spouse might qualify for marriage car insurance discount.