Here are some of the most popular melodies for Sim Shalom:

• Here's a folk tune that I heard used for Sim Shalom in Conservative synagogues in northeastern New Jersey in the 1960s. The prayer's first line, used as the repeating chorus, was interjected by the congregation. The verses were sung by the cantor.
• Here's a faster Sim Shalom tune, currently used in orthodox synagogues in the Boston area. It sounds chassidic to me, but I'll post more about its origins when I figure that out.
• Here's another that's currently popular in Boston area orthodox shuls.

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Sim Shalom (Grant Peace)

It is important to note that the Jewish notion of peace differs fromthe usual Western conception. The English word "peace" comes from the Latin word "pax". "Pax" literally means quiet. The Hebrew word "shalom" comes from the root "shalem," which means "complete" or "whole." While one of the meanings of "shalom" is trans­lated as "peace," the vision expressed in the Hebrew is not the same. "Shalom" means "wholeness," "completeness" or "fulfilledness"; well-being. This is a dynamic conception of "peace." [i] In many contexts, shalom can be taken as a reference to the utopian future, the completed creation...

... In the ancient Temple,[ii] the final element of the service was the Priestly Blessing over the assembled people.[iii] This is called "Birchat Kohanim" in Hebrew. The climactic final thought in the Temple was for "peace." In essence, the people were "dismissed" with the words of Numbers 6:24-26.

In the morning Amidah, the last prayer is Sim Shalom, which is an elaboration and expansion of the orig­inal Priestly Blessing.[iv] Sim Shalom recasts it into the formal structure of a fixed benediction, thereby rendering it appropriate for an individual or reader in the absence of an active priesthood...

... When the sacrifice was first replaced with the Amidah, the end of the Amidah was the end of the whole service. So Sim Shalom was the end of the service. This climactic regard for "peace" is typical. As we shall see later, the kaddish and the Torah-Reading section of the service build to a similar conclusion. (So does the Grace After Meals).

The last benediction has two forms, "Sim Shalom" (Grant Peace) and "Shalom Rav" (Great Peace). Sim Shalom is used in Shacharit and Musaf. Shalom Rav is used for Mincha and Ma-ariv. This is another example of rabbinic compromise. (Some Sephardic communities have never adopted Shalom Rav).

Since many Jews only go to synagogue on Friday night (Ma-ariv, when the Amidah is not repeated aloud) and Sat­urday morning (Shacharit and Musaf), they only hear Sim Shalom. The exceptions are Reform Jews, who have a very lovely modern melody[iv] for Shalom Rav that is often sung aloud on Friday nights.

--- adapted from "The Synagogue Survival Kit" by Jordan Lee Wagner, publ. by Rowman & Littlefield. 1997.