Here is a Video from middle of November it shows where ISIS and Kurdish units are located. Basically ISIS controls the west, south and east of Kobani. Early in December some fire also came from the Turkish boarder (north) side of Kobani
On October 5, ISIS forces captured the southeastern hill overlooking Kobani and breached the town entrance below it (see full article) as shells rained down inside the town. Fresh U.S. airstrikes the same day appeared to have almost no effect. Syrian Kurdish YPG/YPJ resistance fighters continue to fiercely defend every meter of land, as block-by-block fighting began on the southeast. On the other side of the town, ISIS siege forces continue to tighten the noose and are still pressing up the Aleppo Road to the southwest entrance.
Kobani: A brief history – Kobani started out in 1912 as a stop on the Konya-Baghdad railway and was populated by Armenian refugees fleeing the forces of the Ottoman Empire in 1915. The name “Kobani” may be a corruption of the word “company”, although in Arabic the town is called Ayn al-Arab or “the spring of the Arabs”.