What's a quick-and-dirty DIY solution for detecting slow air leaks?
If you take the tire off and just visually inspect it prior to putting soap on it, you'll most likely find the culprit (ie: nail, screw, etc).
If you don't find a "smoking gun", mix a solution of a small amount of dish washing liquid into water. Maybe 20-30:1 (liquid to water)? It doesn't take a large amount of the soap for it to create bubbles. Take the offending tire off of the car and liberally spray an area of the tire. Soak one area of the tire, look and watch for bubbles, then move on to the next area. Ensure you do the sides at the bead as well as the valve stem.
Would I apply it with a sprayer?
Yes. I'd prefer a fine mist over a stream, but either should work just fine.
And, while I'm at it, what kind of bubble production would constitute a "small" leak; a 1/2" bubble per second? a 1/4" bubble per minute?
Any noticeable leak is a "big" leak. IOW, any amount of formed bubbles is too much. I'd suggest if your leak is as slow as you're suggesting, there won't be a huge bubble factory going on, so you'll need to pay close attention.