David E. Cooper argues they’re not just for the religious:
An epiphany does not have to be a theophany, a manifestation of a divine being. In James Joyce’s A Portrait of the Artist as a Young Man, Stephen explains to a friend, as they walk down Westmoreland Street, that a clock on a building is not simply part of “Dublin’s street furniture.” It is an “epiphany,” “a sudden spiritual manifestation,” in which the “whatness” or essence of a great city shines forth. So “epiphany” need not be invested with a theological sense. Experiences of sunlit seas need not have a specifically religious significance. Nor need an epiphany be dramatic and sudden, like Krishna’s appearance before Arjuna. The experience may be relaxed and cheerful: like Stephen’s on a Dublin street, it may be abiding and steady.
What should be retained from the ancient notion of epiphany is the sense of a showing forth — a becoming manifest to experience — of what has been occluded or recessive. We should retain as well the sense that what is manifested is something that matters — a significant, even fundamental, aspect of reality. You can appropriately speak of epiphanies of divine splendor, spiritual force, or the soul of a city, but not, usually, of epiphanies of cabbage leaves and rubber bands. An epiphany is something that is experienced by a person as a showing or bodying forth of a profound aspect of the way of things. An epiphany brings a truth about the world into the sphere of vivid personal experience.
(Image from Camille Flammarion, L’Atmosphere: Météorologie Populaire, 1888, via Wikimedia Commons)