Ahsan Allahu ilaykum. Most women in my country cover their hair using a regular scarf and they wrap it around their neck. The better ones also cover their shoulders and let it fall over their chest. Is this permissible or is it obligatory to wear a khimar/jilbab style, ie something tied on the head?

It is obligatory upon you to wear the Islamic veil as is described in the Quran and is known all over the world. As for what you have described, I need further explanation or maybe to see it (without women showing, of course) in a picture on a mannequin, or just as a standalone image to get a better idea. However, the people of Ilm (the scholars) have said that anything that would show the neck, the shape of the neck, the shoulder, and so on, is prohibited and should not be worn. Therefore, we stick to the proper Islamic hijab and we should not change anything in regards to it.