I understand that A"H is translates to "may he/she rest in peace," which is indeed closer to what non-Jews say in gentile societies. However, I also understand Z"L is also an honorific for the dead, which translates to "may his/her memory be a blessing" or "of blessed memory." Which is more common among Jews? Which is more proper, and for what occasions?
Great rabbis and other figures -- other than Moses, see below -- tend to get zichrono livracha (or zecher tzadik livracha). Alav hashalom is something you'll hear colloquially about a parent or something who wasn't necessarily a great scholar. (Though it's still acceptable to reverse those.) And then there's Moses, for whom the phrase alav hashalom was coined!
Alav haShalom actually arrived at Hebrew via the Arabic! Islam refers to their central prophet as "peace [and blessings] be upon him", so when Maimonides (who lived in Muslim lands almost all his life) wrote in Judeo-Arabic, he wrote "Moses our Teacher -- peace be upon him!" Those works were later translated into Hebrew, and the phrase alav hashalom stuck.