The attacks came after the Islamic State group suffered several setbacks in northern Syria against Kurdish forces over the past weeks.
In Kobani, which famously resisted a months-long assault by the Islamic militants before driving them out in January, an activist group said 10 people died in fighting today the first time in six months the IS had managed to enter the town along the Syria-Turkey border.
Hassakeh is divided between Syrian President Bashar Assad's forces and Kurdish fighters.
Syrian state TV reported intense clashes inside Hassakeh's southern neighborhood of Nashawi. According to the report, IS fighters killed several people they captured in the city, including the head of a military housing institution.
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It said the militants sustained many casualties, including the commander of the group who is a foreign fighter.
IS tried to storm the city earlier this month and reached its southern outskirts before facing strong resistance from Syrian government troops who pushed them away.
Kurdish fighters have been advancing since January under the cover of airstrikes by the US-led coalition.
But in neighboring Iraq, government forces and allied Shiite militiamen have been slow in retaking IS-held territory. The Iraqis have also suffered occasional losses.
Iraqi troops drove IS militants from Saddam Hussein's hometown of Tikrit in April, but lost Ramadi, the capital of Anbar province west of Baghdad, last month.
In an audio message Tuesday, IS spokesman, Abu Muhammad al-Adnani, urged Sunni Muslims to use the time of piety and dawn-to-dusk fasting during Ramadan to wage jihad and seek martyrdom.
"Attack them everywhere and shake the ground beneath them," he said. It was not possible to verify the recording, but it resembled previous audio statements from the group.