Beginning in 2009, the spouse of a servicemember is no longer required to release their residency of the servicemember's domicile state. This means that if the couple was domiciled in State A and then moved (due to the servicemember's military orders) to State B, the spouse can now elect (subject to state instructions, which do vary) to remain a resident of State A as the military servicemember has always been able to.

Per Federation of Tax Administrators, Summary of S.475 Military Spouses Residency Relief Act:

For the sake of simplicity, this document refers to spouses as “she” and servicemembers as “he.”

1. Normally a worker will be taxed by the state in which income is earned. Federal laws now changes the point of taxation for the spouse and the servicemember to the state of domicile.
2. Under longstanding federal law, a Servicemember with a domicile in a state other than where he is stationed can’t be taxed on military income earned in that state. However, the Servicemember still can be taxed by the state on non-military income earned in that jurisdiction. Thus, even though it seems counterintuitive, this exemption from state income tax for spouses is broader than the exemption servicemembers themselves receive.
3. A spouse who moves to a new state, establishes a new residence and a new life there, would normally become a domiciliary of that new state. The Act allows a servicemember’s spouse to keep a previous domicile (under qualifying conditions enumerated below). The spouse may, however, choose to become a domiciliary of the new state.
4. Eligibility requires three factors. The spouse of a servicemember is exempt from income taxation by a state when she:
   (1) Currently resides in a state different than the state of her domicile;
   (2) Resides in the state solely in order to live with the servicemember; and,
   (3) The servicemember is present in the state in compliance with military orders.
5. Eligibility may require four factors in some states. See Item 12 for discussion of this requirement:
   (4) The spouse and the servicemember both are able to claim the same domicile.

[Refer to the previous link for additional details about the Military Spouses Residency Relief Act.]

Per the original Servicemembers Civil Relief Act ("SCRA") (Original Bill), starting on page 34:

§ 571. Residence for tax purposes

(a) Residence or domicile.
         (1) In general. A servicemember shall neither lose nor acquire a residence or domicile for purposes of taxation with respect to the person, personal property, or income of the servicemember by reason of being absent or present in any tax jurisdiction of the United States solely in compliance with military orders.
         (2) Spouses. A spouse of a servicemember shall neither lose nor acquire a residence or domicile for purposes of taxation with respect to the person, personal property, or income of the spouse by reason of being absent or present in any tax jurisdiction of the United States solely to be with the servicemember in compliance with the servicemember’s military orders if the residence or domicile, as the case may be, is the same for the servicemember and the spouse.
(b) Military service compensation. Compensation of a servicemember for military service shall not be deemed to be income for services performed or from sources within a tax jurisdiction of the United States if the servicemember is not a resident or domiciliary of the jurisdiction in which the servicemember is serving in compliance with military orders.
(c) Income of a military spouse. Income for services performed by the spouse of a servicemember shall not be deemed to be income for services performed or from sources within a tax jurisdiction of the United States if the spouse is not a resident or domiciliary of the jurisdiction in which the income is earned because the spouse is in the jurisdiction solely to be with the servicemember serving in compliance with military orders.

Per the Military Spouses Residency Relief Act (Revised Bill) Determination for Tax Purposes of Residence of Spouses of Military Personnel, Section 3, page 2:

(1) IN GENERAL.—A servicemember; and
(B) by adding at the end the following:
(2) SPOUSES.—A spouse of a servicemember shall neither lose nor acquire a residence or domicile for purposes of taxation with respect to the person, personal property, or income of the spouse by reason of being absent or present in any tax jurisdiction of the United States solely to be with the servicemember in compliance with the servicemember’s military orders if the residence or domicile, as the case may be, is the same for the servicemember and the spouse.

(c) INCOME OF A MILITARY SPOUSE.—Income for services performed by the spouse of a servicemember shall not be deemed to be income for services performed or from sources within a tax jurisdiction of the United States if the spouse is not a resident or domiciliary of the jurisdiction in which the income is earned because the spouse is in the jurisdiction solely to be with the servicemember serving in compliance with military orders.

Additional Information

Spouses claiming exemption from their "non-domicile" state income tax should consider the impact on their income tax (and estimated income tax) liability in their domicile state. Any income the military spouse earns in a jurisdiction will not be treated as income from services performed or sources within that jurisdiction if that spouse is not treated as a resident of the jurisdiction under the act.

Note that any link in the information above is updated each year automatically and will take you to the most recent version of the document at the time it is accessed.