* Islamic State sets sights on Iraq's oil fields
* Militants inflict first major defeat on Kurds
* Political paralysis helps Sunni insurgents
By Ahmed Rasheed and Raheem Salman
BAGHDAD, Aug 3 (Reuters) - Islamic State Sunni insurgents have captured two northern Iraqi towns and an oil field in their first major victory over Kurdish fighters, witnesses said on Sunday.
The al-Qaeda offshoot, which swept through northern Iraq in June almost unopposed by Iraq's U.S.-trained army, poses the biggest challenge to the stability of the country since the fall of Saddam Hussein in 2003.
After thousands of Iraqi soldiers fled the Islamic State offensive, Shi'ite militias and Kurdish fighters have emerged as a key line of defence against the militants, who have threatened to march on Baghdad.
Kurdish forces poured in reinforcements, including special forces, to the town of Zumar this weekend to battle Islamic State fighters who had arrived from three directions on pickup trucks mounted with weapons, residents said.
The militants later hoisted their black flag over buildings in Zumar, a ritual that has in the past been followed by the mass execution of captured opponents and the violent imposition of an ideology that even al-Qaeda finds excessive.
The Islamic State later also seized the town of Sinjar, where witnesses said residents had fled after Kurdish fighters put up little resistance against the militants.
Islamic State has stalled in its drive to reach Baghdad, halting just north of the town of Samarra, 100 km (62 miles) north of the capital.
The Islamic State in Iraq and the Levant (ISIL) changed its name earlier this year and declared a caliphate in parts of Iraq and Syria. The group has already seized four oil fields, which help fund its operations.
The group has been trying to consolidate its gains, setting its sights on strategic towns near oil fields, as well as border crossings with Syria so that it can move easily back and forth and transport supplies.
It has capitalised on sectarian tensions and disenchantment with Iraq's Shi'ite Prime Minister Nuri al-Maliki.
Critics describe Maliki as an authoritarian leader who has put allies from the Shi'ite majority in key military and government positions at the expense of Sunnis, driving a growing number of the religious minority in Iraq to support the Islamic State and other insurgents. He is also at odds with the Kurds.
The Kurds have long dreamed of their own independent state, an aspiration that has angered Maliki, who has frequently clashed with the non-Arabs over budgets, land and oil.
After the Islamic State arrived, Kurdish forces seized two oil fields in northern Iraq and took over operations from a state-run oil company.
In July, the Kurdish political bloc ended participation in Iraq's national government in protest over Maliki's accusation that Kurds were allowing "terrorists" to stay in Arbil, capital of their semi-autonomous region known as Kurdistan.
In another move certain to infuriate the Baghdad government, the Kurdish region is pressing Washington for sophisticated weapons it says Kurdish fighters need to push back the Islamist militants, Kurdish and U.S. officials said.
But Maliki needs the Kurds to help defend his government.
Maliki is currently ruling in a caretaker capacity, having won a parliamentary election in April but failing to win enough support from the Kurdish and Arab Sunni minorities as well as fellow Shi'ites to form a new government.
He has rejected calls by Sunnis, Kurds and Shi'ites to step aside so a less polarising figure can form a government capable of easing sectarian tensions and countering the insurgency.
An official in the Northern Oil Company said Islamic State fighters had taken control of the Ain Zalah oil field and two other undeveloped fields - Batma and Sufaiya.
In a statement on its website on Sunday, Islamic State said its fighters had killed scores of Kurdish fighters in a 24-hour battle and then had taken over Zumar and 12 villages.
"Hundreds fled leaving vehicles and a huge number of weapons and munitions and the brothers control many areas," Islamic State said. "The fighters arrived in the border triangle between Iraq, Syria and Turkey," it said.
Islamic State's ambitions have alarmed other Arab states who fear their success could embolden militants in their countries.
Eight Lebanese soldiers were killed in clashes with Islamist militants that began on Saturday in and around the town of Arsal near the Syrian border and continued overnight, the army said.
Earlier Lebanese security officials said at least 11 militants and three civilians had been killed in the fighting there and that around 16 members of the security forces had been taken hostage.
The militants included fighters from al Qaeda's Syrian branch and from Islamic State.
On Friday, Saudi Arabia's King Abdullah called on regional leaders and religious scholars to prevent Islam from being hijacked by militants. (Writing by Michael Georgy; Editing by Janet Lawrence and Gareth Jones)