Regarding kedushah desidra after shacharit, it would appear that there are different customs.
Aruch HaShulchan 132:8 writes that he believes it to be appropriate to sit throughout ashrei - uva letzion. Similarly, Kaf HaChaim 59:20 writes, based mostly on kabbalistic sources, that the kedushah of uva letzion should be recited sitting and not standing.
On the other hand, R. Wosner writes in Shevet HaLevi VI:13 that he hasn't seen much explicit on the subject, but he believes that the custom is to stand, and he acts thus himself. He points out that such practice certainly makes sense if the reason for reciting this kedushah is as a substitute for the kedushah of the amidah. (See Beit Yosef, Orach Chaim 132:2.) He concludes that either sitting or standing is acceptable.
Regarding kedushah desidra on motzaei shabbat, Piskei Teshuvot Orach Chaim 295:1 points out that this is not being said as a kedushah at all, as there is no kedushah recited at arvit. Rather, we are merely prolonging the service, and thereby delaying the return of the wicked to gehinnom (see Tur, Orach Chaim 295). Therefore, he believes that one can recite this kedushah in whatever postion one likes.
In a footnote, he points to a comment of R. Yitzhak ibn Ghayyat (quoted in Biur Halachah to 489:1) which implies that the custom in his day was to sit. He also notes a line from R. Yaakov Tzemach's Nagid uMetzaveh which sounds like one should specifically sit, presumably for kabbalistic reasons.
However, he writes that he believes that the common custom is indeed to stand. He conjectures that the reason people do this is because they have the practice to stand for vihi noam and Tehillim 91 (see e.g. Sha'arei Teshuvah 295:1). They therefore simply remain standing until the end of the service.