In every community I have been in there is a custom to sing Lecha Dodi, sometimes along with yedid nefesh or shir hashirim, during kabbalat shabbat. Are there any communities which do not sing lecha dodi but something else instead? Before the song was composed was there a tradition of singing a song during kabbalat shabbat or did they simply go from kabbalat shabbat directly into maariv?
Minhag Beit El and the various Yeshivot HaMekubalim which have grown out from them over the past 300+ years do not sing Lecha Dodi. The Minhag is recorded in the book Divrei Shalom, by Rav Raphael Sharabi(grandson of the Rashash) in the Minhagei Beit El section #37. There he gives two reasons 1)The Ari did not explicitly tell his students to sing it, 2) The Ari specifically forbade the addition of piyutim written after the period of Geonim to the prayer services.
One specific source is Shaar HaKavvanot Drush Nusach Tefilot p 50c:
Thus we find that the song is conspicuously absent from the both the Siddur of his student, Rav Abulafia, as well as the siddur that the Rashash composed for the Hakhamei Tunis that was published in the Aram Tzova version.
Also the sefer Nekudot HaKesef goes a bit into depth on the matter of singing the song and whether it poses a problem either halakhically or kabbalistically.