To answer your question, the choice of which one to use is left pretty much to the priest-celebrant’s discretion. There are, however, in No. 365 of the General Instruction of the Roman Missal, some guidelines that help the priest decide which prayer might be most appropriate — with respect, at least, to the four basic options.Here's a post by Fr Barbour at Catholic Answers:The Eucharistic Prayer 1 (sometimes referred to by its former title, the “Roman Canon”) is especially appropriate on major feasts, since it provides for references to that feast to be included in the prayer itself. It is also suitable on feast days of those saints who are mentioned by name in the prayer.
Eucharistic Prayer 2 is the briefest of the four options and for that reason is often used for weekday Masses.
Eucharistic Prayer 3 is “preferred on Sundays and festive days,” and Eucharistic Prayer 4, which is the lengthiest of the four, “gives a fuller summary of salvation history.” (I tend to use this one when I am celebrating with a congregation that is especially in touch with biblical theology or, sometimes, as a change of pace with a weekday congregation.)
There is no strict rule regarding the choice of the eucharistic prayer at Mass, except for Eucharistic Prayer IV, which may be used only on days that do not have a strictly proper preface.I m a fairly new Catholic, and my ear isn't trained for nuances, but I'd guess that very nearly every one of the 350 or so Sunday or day-of-obligation masses I've attended in three dioceses has been Eucharistic Prayer III. Certainly I've noticed no jarring differences among them. And this suggests that although visitors may disagree with the premises of Vatican II or its fallout and feel that the Roman Canon should be the one in normal use, given what seems to be a consensus in the real-world US Church, these dozens of priests are simply doing what's licit and generally practiced.. . . Eucharistic Prayer III is suitable especially for Sundays, and, I add, it conveniently has a place for the commemoration of the saint of the day or of the patron of the place or community.
Also, as I've pointed out here, as a convert, it's not my role to fix the Church. If I thought there was something wrong with it, I could have remained a Protestant, after all.
Since I compared EP III to the Divine Worship canon, I was in fact comparing Sunday mass to Sunday mass. One thing that puzzles me is that for those who prefer EP I, why isn't there a much larger movement for parishes to celebrate EP I? There's a Latin mass movement, which has some traction -- certainly more than the ordinariates -- but although I haven't looked for one, nobody's been in my face with "our parish celebrates EP I every Sunday!" or even, like the Episcopalians, Rite Two at 10:30, Rite One at 8:00.
And that leaves open the question of why anyone would want EP I in emended archaic language plus the extra Cranmerian prayers.
I'm also told that Bp Lopes in his talk to the Toronto conference last November mentioned, as something of a joke, that he had a problem with priests who didn't understand that the multiple options in the DW mass were or, not and. Funny, huh? But that's what you get with all those ordinations that amount to 30 seconds in the microwave.