I. The Styles
Certain sections of the prayer service are commonly sung by the congregation but there are two main ways in which this singing takes place: 1) the Beis Medrash style and 2) the Young Israel style.
In the former, the leader sings a verse and then after him the congregation recites it. Then the leader sings another verse and after him the congregation recites it. Etc. In many synagogues, while the leader is singing everyone sings along without actually saying the words, perhaps with "ay yay yay" and waiting until the congregation's turn to recite rather than sing the words.
In the Young Israel style, the leader and the congregation all sing the verses together straight through. It is my belief, and for what little a vague second-hand recollection is worth I think I heard that Rav Soloveitchik said similarly, that the Beis Medrash style is what Ashkenazic Jews have traditionally done and the Young Israel style was an intentional break with tradition.
II. The History of Young Israel Style
In the early 1910's, the Young Israel movement was founded in order to keep young Jews within Orthodoxy. Synagogues were considered uninviting and uninteresting to many of these young Jews, so Young Israel attempted to reinvigorate the synagogue experience. Part of this included the adoption of certain non-Orthodox practices that were acceptable within Jewish law, even if they were themselves likely adopted from Christians. Primary among them was congregational singing -- everyone together -- which is more participatory than merely
listening to a cantor sing.
I don't have irrefutable proof that this happened but there are some semi-historical books that either say this explicitly or hint to it:
1. R. Berel Wein, Triumph of Survival, p. 335:
Young Israel had become a national movement with branches over the entire country. It encouraged youth programs, communal prayer with song, English-speaking rabbis, and strong commitment to tradition coupled with a positive attitude toward American life and society.2. Victor B. Geller, Orthodoxy Awakens, p. 152:
The growing Young Israel movement was a thoroughly American phenomenon. Established in 1912 on the Lower East Side of New York, it was a moderate "in-house" revolt of observant children of Orthodox immigrants. While adhering to Torah practices, these young people felt the need to address some of the existing practices in local Orthodox synagogues. They established a new synagogue (it continues today as the Young Israel of Manhattan), which discontinued the practice of demanding "contributions" for aliyot (being called to the Torah). They also insisted on proper synagogue decorum and instituted congregational singing.3. R. Aryeh Leib Scheinbaum, The World That Was: America 1900-1945, p. 19:
Furthermore, many of the younger generation also objected to the sale of honors to the wealthy, and to services completely devoid of melodies and congregational participation. The youth, torn between their parents' worldview and their own, devised a solution which, despite its shortcomings, helped save a generation of Orthodox Jewish youth. In all likelihood, without Young Israel thousands of Jewish youth might bever have remained Torah observant, a fact many of us today tend to forget.Whether or not this break with tradition was permissible is a long discussion. See this post of mine for a discussion of this issue. I prefer the Beis Medrash style because, historically, it is more authentic. However, I don't think that anyone can deny that the Young Israel style was begun with proper intentions and can be continued. What I do object to, however, is when people in a synagogue that follows one practice choose to adopt the other. For example, Young Israel members who decide that their synagogue's practice is not "frum" enough for them and try to unilaterally change it are, it seems to me, deviating from the synagogue's custom. The same applies to something that I have seen a few times in Beis Medrash style synagogues, particularly when the rabbi is away, that certain members try to get everyone to sing along with the leader. This, too, it seems to me, is deviating from the synagogue's practice, and in particular towards a less authentic practice.
III. The Talmudic Precedent
I believe that there is a Talmudic basis for the Beis Medrash style singing and, according to one interpretation, the Young Israel style as well. The Gemara (Sotah 30b) records three opinions about how the Israelites fleeing Egypt sang the song after the splitting of the Reed Sea. According to R. Akiva, Moshe sang (or recited) the song verse by verse and after each verse the nation repeated only the very first phrase of the song. According to R. Eliezer the son of R. Yossi from Galilee, Moshe sang the song verse by verse and the nation repeated each phrase after him. And according to R. Nechemiah, they all sang together (as Rashi explains his position, but see Tosafos who explain differently).
If this means that Moshe and the nation actually sang the song, rather than merely reciting it (perhaps to background instrumental music), then R. Eliezer's position matches the Beis Medrash style of singing and R. Nechemiah's position, according to Rashi, matches the Young Israel style.