I asked Is it OK to compose a song that breaks up a pasuk? and was told (in comments) that perhaps it would be good to first clarify whether songs can use pesukim at all.
So, question: Can songs, general Jewish music, use words from Tanach (and other places), or has the entire Jewish music industry of the past ~60 years been against halacha?
תנו רבנן הקורא פסוק של שיר השירים ועושה אותו כמין זמר והקורא פסוק בבית משתאות בלא זמנו מביא רעה לעולם מפני שהתורה חוגרת שק ועומדת לפני הקב"ה ואומרת לפניו רבונו של עולם עשאוני בניך ככנור שמנגנין בו לצים
ה"ג הקורא שיר השירים ועושה אותו כמין זמר. שקורא בנגינה אחרת שאינו נקוד בה ועושה אותה כמין שיר אע"פ שמשיר השירים הוא ועיקרו שיר אסור לעשותו כמין שיר אלא בקריאתו:Our Rabbis taught: He who recites a verse of the Song of Songs and treats it as a [secular] air, and one who recites a verse at the banqueting table unseasonably, brings evil upon the world. Because the Torah girds itself in sackcloth, and stands before the Holy One, blessed be He, and laments before Him, 'Sovereign of the Universe! Thy children have made me as a harp upon which they frivolously play.'
That he reads it with a different tune than what it is pointed with and makes it like a song, even though it is from the Song of Songs which is essentially a song, it is forbidden to make it like a song except in its [original] way. (Soncino translation of Talmud; my translation of Rashi)
Rav Eliezer Waldenberg (Tzitz Eliezer 22:10) explains the prohibition does not apply if the songs are directed as praise to God. Rav Moshe Feinstein (Igrot Moshe YD 1:173) writes that it is forbidden to make recordings of verses being sung for the purpose of listening to them for pleasure. Rav Ovadia Yosef (Yabia Omer OC 3:15) writes that the prohibition does not apply if the songs are sung with fear of heaven and "level-headedness" (כובד ראש) at a celebration of a Mitzva. Rav Yaakov Ariel writes that it is only permitted to sing verses if one focuses on their content and on one's connection with God. Rav Yosef Shalom Elyashiv writes that one can listen to such songs only with "holy trepidation" (חרדת קודש).
The Mishna Berura (560 sk 14) quotes early authorities that this applies to any verse in Tanakh, not just Shir HaShirim. Rav Moshe Feinstein (Igrot Moshe YD 2:142) speculates that perhaps some are lenient in this matter by viewing the prohibition as referring specifically to Shir HaShirim (which is prone to being misinterpreted as "mere song") though he doesn't know of any early authority to support this and leaves the matter unresolved (צ"ע).
In short, Tanakh is not just a cute book of playful lyrics for your own relaxation and enjoyment. It is very holy and its use in song is only justified as a bona fide religious experience.
This answer is a summary of Rabbi Jachter's writeup on this subject.
He provides four (and a half) justifications for why putting pesukim to music is permissible.
The first is that the prohibition was only for Shir Hashirim, because if it is put to music, it is more prone to being misinterpreted as a simple love song. (suggested but not accepted by Igrot Moshe YD 2:142; 1963) The article says this is also implied in Avot D'Rabi Natan (perek 37), but i personally didn't see it.
The second justification is that it is only assur if it's done as a mockery, even if it's not being done as a mitzvah (m'sameach chatan v'kallah). Therefore, most cases should be permissible. Rav Yaakov Emden seems to think along these lines in his commentary on the passage in Sanhedrin.
The third justification (brought unsourced) is that in today's world, when non-Jewish music is so readily available and popular, to davka listen to Jewish music helps enhance your connection to Hashem. Therefore, we can even consider this to be for the purpose of a mitzvah!
The fourth (also brought unsourced) is simple survival as an Orthodox Jewish community. Now that we've moved out of the shtetls, non-Jewish influences are much stronger, and having a strong cultural identity -- in addition to religious practices -- help us preserve our lifestyle.
Also, he writes, פוק חזי -- go see what the common practice is. Because the lenient opinion is so popular, it must be correct (based on Ramban Devarim 17:11).
If you wish to be machmir on yourself and not listen to Jewish (or any) music, that's also permissible, but for me, bring it on!
