The Ben Ish Hai (Rabbi Yosef Haim of Baghdad, 1833-1909), in Parashat Vayehi, addresses the issue of standing during Kaddish (listen to audio recording for precise citation). He writes that if one was standing when Kaddish began, then he must remain standing until the end of "Yeheh Shemeh Rabba Mebarach…" For example, when the Hazzan recites Kaddish after the Amida or after Hallel, the congregation is already standing, and they should therefore remain standing until after the response of "Yeheh Shemeh Rabba." If a person had been sitting when Kaddish began, he may, according to Sephardic custom, remain seated. (This is contrast to the practice of the Ashkenazim, who always stand for Kaddish.)
We should add that this Halacha applies also on Friday night, when the Hazzan recites Kaddish after the "Me’en Sheba." Unfortunately, many people rush to sit down before the Hazzan begins Kaddish, so that they could remain seated during the Kaddish. This is improper. Since they had already been standing, they should remain standing until at least the end of the response of "Yeheh Shemeh Rabba."
The Ben Ish Hai then proceeds to caution against reciting Kaddish unnecessarily. Just as Halacha discourages the recitation of a "Beracha She’ena Sericha" – a Beracha that one did not have to recite – similarly, a Minyan should not recite Kaddish simply for the sake of reciting Kaddish. The exception to this rule, the Ben Ish Hai notes, is reciting Kaddish after learning Torah. After a group studies Humash or Tanach, they recite "Kaddish Yeheh Shelama," and after studying Torah She’be’al Pe (the Oral Law – Mishna or Gemara), they recite "Kaddish De’Rabbanan." The Ben Ish Hai writes that in Baghdad, after groups would finish studying Torah She’be’al Pe and recite "Kaddish De’Rabbanan," they would then recite some Tehillim followed by "Kaddish Yeheh Shelama." This is our practice, as well, on the night of Shabuot and the night of Hoshana Rabba, and this is certainly acceptable. In other contexts, however, it is inappropriate to conduct extra Kaddish recitations unnecessarily.
Summary: One who was standing when the Hazzan began reciting Kaddish must remain standing until the end of "Yeheh Shemeh Rabba Mebarach"; otherwise, he may sit during Kaddish. A congregation should not recite Kaddish unnecessarily, though they may study or recite Tehillim and the like in order to then recite "Kaddish Yeheh Shelama," as is customarily done on Shabuot and Hoshana Rabba.