There is a variety of placement in siddurim which include Askinu Seudasa and Azamer Bishvachin. Some (Tefila Yeshara, Artscroll if memory serves correctly) put both before Kiddush. Some (Tehilas Hashem) put Askinu Seudasa before Kiddush and Azamer Bishvachin is placed after Hamotzi as a zemer. I have also heard that some say both as zmiros, although I haven't seen a siddur which evidences this practice.
There also seems to sometimes be a variance in practice between evening Kiddush and morning Kiddush.
What are the reasons behind the differing placements? Which groups follow which opinion?