I'm not too fond of the adjective "spiritual" but I can relate to the kind of experiences described thus far. Often causing a definite shudder in my spine, or making me slow down suddenly if I am walking.
"Nature," yes. A warm sunbeam in the foliage of a tree in the evening. A sudden breeze in the forest, on the fields, on water (thinking of Tarkovsky here). Snow of course.
GF mentioned reality and it brings back one experience: the first time I saw Saturn on a telescope. I had seen dozens of far better pictures of it and "knew" it was there but at this moment I realised it was there and I was there... Hours later this impression was still on my mind.
However it is not limited to nature, culture makes it too: music, of course; paintings, sometimes; movies, often. Or just an unexpected "right" word or phrase coming up in a conversation, in a book, or in my own mind.
On the "religious" side, I remember once praying (really praying, not the JW parrotting) with a very dear friend when I was still in Bethel. I suddenly had the strange impression of a modification of space, as if the room we were sitting apart in had become immense and we were filling it and there was no distance between us. A very odd feeling.
My favourite word for such "experiences" is ecstasy. Being for an instant as if out of oneself. To me they don't prove anything, and are probably not above explanations (neurological, for instance). But this in no way reduces their subjective force.