Question: I never got a clear answer as to whether, on Motzaei Shabbat, we light Chanuka candles before or after Havdala. Can you clarify the matter?

Answer: In terms of practice, we can clarify only a few things. The minhag in shuls is to light Chanuka candles first (based on Terumat Hadeshen 60, Shulchan Aruch and Rama, Orach Chayim 681:2). Regarding the home, Sephardim do Havdala before Chanuka candles (Kaf Hachayim, OC 681:4; Yalkut Yosef; Mikraei Kodesh (Harari), Chanuka 11:10). Among Ashkenazim, some schools of thought have clear rulings (which vary one from another); we will not provide lists of the opinions. However, the standard approach, to which we subscribe is that this is a case where one may follow the approach he wants (see Mishna Berura 681:3). We note that not only are both approaches well grounded, but also the question is only of preference; following the "incorrect" approach is not a "violation." We will survey some of the indications presented by prominent protagonists.
The Terumat Hadeshen’s reason for lighting Chanuka candles first is the gemara’s (Pesachim 105b) concept that we delay Havdala (i.e., put it at the end of the series of berachot) and, thereby, the exit of Shabbat, so that Shabbat not appear as a burden. The Taz (OC 281:1) counters the Terumat Hadeshen’s assumptions on two fronts. First, he argues that tadir kodem – a more common beracha is recited before a less common one – is a stronger factor than delaying the end of Shabbat. Additionally, argues the Taz, doing Chanuka candles before Havdala is not even a correct application of delaying the end of Shabbat. This is because lighting candles itself contradicts the continuation of Shabbat, for if it were still Shabbat, lighting a candle would be forbidden.
The following discussion in Tosafot (Shabbat 23b) can shed light on the relative strength of the factor of tadir kodem. The gemara says that if one has enough money only for Chanuka candles or wine for Kiddush (which is more tadir), Chanuka has precedence because pirsumei nisa (publicizing the miracle) is more important. Tosafot asks why, regarding Rosh Chodesh on Chanuka, we read the haftrara of Chanuka, yet read the Torah portion of Rosh Chodesh before that of Chanuka. Tosafot’s first answer, which the Taz cites as support, is that the advantage of pirsumei nisa prevails when only one of the mitzvot can be fulfilled, whereas tadir kodem is the key factor regarding the order when both are done. However, points out the Eliyah Rabba (681:1), Tosafot’s other two answers diminish the importance of pirsumei nisa only regarding the Torah/haftara readings. This implies that when pirsumei nisa applies, as it does to Chanuka candles, it has order precedence over the more common Havdala.
R. Yaakov Emden (Mor U’ktzi’ah 681) rejects the Taz’s claim that lighting Chanuka candles contradicts the idea of delaying Havdala/end of Shabbat. He points out that after Havdala in davening or Hamavdil, we are allowed to do work (including Yom Tov candles and Borei Me’ore Ha’eish) before doing Havdala over wine, and yet we delay the beracha of Havdala.
Some cite a proof that Chanuka lighting precedes Havdala from the Yerushalmi (cited by the Shulchan Aruch, OC 581:1), that one should not use the Chanuka candles for Borei Me’ore Ha’eish. This implies that the Chanuka candles are lit first.
Is there any logic, other than minhag, to switch the orders in shul and at home, as Sephardim and some Ashkenazim do? The Maharal (Ner Mitzva, p. 28) cites, as a reason to do Havdala first, the concern that one who one did not say Havdala in Shemoneh Esrei will light in violation of Shabbat. The Eliya Rabba points out that in shul, we trust that the person appointed to light will be a diligent person who will not forget. It is also possible that since pirsumei nisa is a heightened element and exists for a shorter time in shul, we do it as soon as possible. The Kaf Hachayim (681:4) claims that after men have heard Havdala in shul, delaying the one at home is less important.