Which denominations believe in modern angelic interventions?

Most denominations that I know of believe that angels are always involved in human affairs without being noticed by anyone (Heb 31:2) and are almost never physically manifesting themselves to people. Even in the Bible there are many prophets, apostles and great men of faith that have no record of an angelic physical manifestation. Of course, at the time of Christ and during the lives of the Apostles there is a peak occurrence of, not only angelic manifestations, but demonic possessions recorded in the Bible.

Thus it happens to many a man that he will escape fire, water, murder, and other misfortunes because of some insignificant thing that has moved him. Such a thought, or whatever it may be, comes to him so that he does, all at once, the thing that saves him, a thing which he could never have foreseen or thought beforehand but of which he must say, “Indeed, if I had done this or that, I surely should have drowned, been burned to death or murdered, or died or suffered harm in some other way”; or, as someone else might say, “You had your guardian angel with you there.” The Gentiles therefore ascribed such happenings to good fortune and made an idol of fortune. For they saw and learned that such things happened but did not know that the true God had done them through His holy angels. So it happened that St. Augustine, when the heretics were lying in wait to kill him, took a different road without any further thought but undoubtedly because of his angel’s doing. And when Emperor Julius leaped from the ship and escaped his enemies by swimming, he surely was of good courage and confidence;11 yet his counsel and courage had been inspired in him by his guardian angel from without and by God from within. Thus it is with all men when they escape misfortune or have good fortune: it is all the work of God and the angels.( Luther, M. (1999). Luther’s works, vol. 20: Minor Prophets III: Zechariah. (J. J. Pelikan, H. C. Oswald, & H. T. Lehmann, Eds.) (Vol. 20, p. 170). Saint Louis: Concordia Publishing House.)

There is no historical denomination over the last two thousand years that urges some sort of recognition that every day Christianity should be somehow more spiritual than the days of Christ, through some odd unexplained manifestation of angelic beings for random purposes. (At least no denomination that I am aware of).

When we look at all the visitations of angels that are in the Bible they almost all seem to be related to spreading the knowledge of the coming Messiah, to directly support the Messiah, or to support the ministry of the Apostles in laying down the foundation of the gospel for the church. There are of course some other instances where angels are involved in judgments of God upon sinners and other miscellaneous interventions.

It only makes sense therefore that if the primary purpose of the angels (with respect to physical representations) is to help promulgate the gospel at key points in the history of the church’s battle against the Devil, individual encounters for the sake of experiences or for some comparatively minor subject, seem not to be expected or to be even claimed by most every believer. I personally have never met a believer that claimed to have an experience with an angel.

To offer a possible answer then to the question of which denominations think angels are running around manifesting themselves to people today, basically none. If asking are angels busy and involved in human day-to-day affairs, I guess about all denominations. If asking is it possible that God could still direct an angels to physically manifest themselves in some important situation for the continued promulgation of the gospel or some other unusual spiritual situation of great importance, I would say again all denominations. Most people do not put a rule on what God can do just because he doesn’t seem to usually.