The music scene in Israel is as diverse as the country's population. From traditional Jewish tunes sung in Ladino to Hebrew Hip-Hop to Israeli Arab music and from Mizrahi songs (Arabic / Mediterranean style) to original Israeli classical works to modern Hebrew poems set to music - you name it.
Music in Israel (Wikipedia): Offers a good introduction to the history and current trends of music in Israel, including "Music and Politics".
Garland encyclopedia of world music (Vol. 6; The Middle East): Includes a number of articles about Israeli music.
Music Collection and Sound Archives (National Library of Israel): Several collections and playlists, including Hasidic niggunim, musical traditions of Yemenite Jews, Jewish holiday songs, and more.
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Hebrew Songs on the Web
Piyut: Old and new liturgical poems; the nonprofit site was recently re-launched at the National Library of Israel and now offers playlists arranged by different Jewish traditions and occasions (holidays, Shabbat, weekdays).
Zemereshet: An online database for early Hebrew songs. Volunteer-based project that includes thousands of recordings from the pre-state days to the 1950s. Informational and scholarly.