• Sumerian cuneiform was undisputedly the first writing system to be invented.
• Egyptian hieroglyphics was probably made with knowledge of cuneiform, but could have been an independent invention. All Western writing systems (Hebrew, Arabic, Greek, Latin etc.) are ultimately descended from hieroglyphics.
• All native writing systems of India, SE Asia and Oceania are descended from Brāhmī. I believe the current theory is that Brāhmī was descended from Aramaic script and hence also from hieroglyphics, but there are disputes.
• Chinese characters (hànzì) were an independent invention. Descended from them are Japanese hanja and kana, Vietnamese Chữ Nôm, Sawndip and some others.
• The writing systems of Mesoamerica (most notably that for Mayan) are of course a completely independent invention.
And then of course there are Hangeul, Cherokee, CAS etc., which were scripts consciously designed under the influence of other scripts. These tend to have rather unique structures compared to other writing systems of the world.