Top Kurdish officials told AFP their fighters were advancing "street by street" and voiced confidence that IS would soon be ejected from the town.
The Syrian Observatory for Human Rights, a Britain-based monitoring group, said the latest advances were mainly in the south of the besieged town.
"The (Kurdish) People's Protection Units (YPG) recaptured streets and buildings in the south of Kobane, after a fierce battle against IS that began yesterday (Monday) evening," the Observatory said.
Kobane has been under siege from IS since mid-September and more than 1,000 people have been killed in the fighting, most of them jihadists.
Syria's Kurds have been backed by Iraqi peshmerga fighters and Syrian rebels in their fight for the town, as well as a wave of air strikes by the US-led coalition against IS.
Syrian Kurdish chief Saleh Muslim said YPG forces were advancing "street by street" and that they would "recapture the town in a very short time".
The Kurds' top field commander in Kobane, Narin Afrin, a 40-year-old woman, said by telephone: "We have been resisting for 56 days in very difficult conditions."
Kobane has become a symbol of resistance against jihadists who control swathes of territory in Syria and Iraq, committing brutal abuses against rivals and the local population.
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