Kobane, also known as Ain al-Arab, has become a strategic battleground between the IS group and its opponents, who include the United States and its Western and Arab allies.
The jihadists launched their latest assault on Kobane yesterday after a three-week siege with a wave of suicide bomb attacks, Mustefa Ebdi, a Kurdish activist from the town, said on his Facebook page.
The Syrian Observatory for Human Rights said the IS militants wrested three areas after day-long battles.
The Observatory's Rami Abdel Rahman earlier said the jihadists and the Kurds were "clashing in the streets, between apartment buildings," sending hundreds of civilians into flight towards the Turkish border.
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Kurdish fighters meanwhile ordered all civilians to evacuate Kobane, Mustafa Bali, a spokesman for Kurds in the city, told AFP, adding that some 2,000 people had left the city.
The IS advances came after an AFP photographer reported seeing two black IS flags flying on Kobane's eastern side, after the jihadists seized at the weekend part of Mishtenur Hill, although the US and its allies tried to slow them with air strikes.
It was the first reported instance of a female Kurdish fighter employing a tactic often used by the jihadists, said the Britain-based monitor, which relies on a network of sources inside war-ravaged Syria for its reports.
The bomber, in her 20s, was a full-time YPG fighter identified as Dilar Gencxemis, alias Arin Mirkan, from Kurdish-controlled Afrin in northwestern Syria.
"She killed dozens of gang members and demonstrated the YPG fighters' determined resistance," her group said.
Kobane has become crucial in the international fight against the jihadists, who sparked further outrage at the weekend with the release of a video showing the beheading of Briton Alan Henning.