I think it must have been during my first trip into NaNoWriMo that I encountered the term "plot bunny", and I've found it immensely useful in describing my thought process. Plot bunnies, for those who haven't been introduced, are those little bits of an idea that spring up - usually when the last thing you're doing is even thinking about writing - and demand your attention. It can be something small, like an image that is just begging for you to write a scene around it, or more complex, like the basis of a civilization. Admittedly, the latter tend to come to me in the form of fever dreams, but they are plot bunnies either way.
I think Terry Pratchett said it best when he said "Inspirations sleet through the universe continuously," and sometimes there's just no getting out of the way. Sometimes I feel like Richard Madoc from Neil Gaiman's Sandman series, constantly hit with ideas. It takes some time, but I've found a decent way to herd the plot bunnies. At least it keeps them from keeping me up at night.
I keep a document on the cloud (so I can access it from work or from my phone, if needed) called Plot Bunnies. Any time one of those ideas comes about, I jot something down to serve as a placeholder for the idea. Sometimes it's just a phrase that caught my attention, sometimes it's a character or a location. Later, when I'm looking for inspiration or need an idea for something I'm writing, I'll peruse the file to see if anything strikes me as useful for that particular scene. It's come in handy a large number of times, even if some of the ideas jotted down make absolutely no sense under the harsh light of day (see also: fever dreams).
I also started a Pinterest board called "Plot Bunny Food" (I'm including the link on the sidebar) that includes photos or links to articles that I think could come in handy or serve as inspiration. I was fortunate to have a member of my writing group who frequently perused the Reddit forums and would send along things she found interesting, so I created the board to help me keep track of it all.
How do you handle random acts of inspiration? Do you have some way of tracking the inspirations as they come sleeting down, or do you let them bounce harmlessly off of you?