A few, though generally polite and positive conversations ensued.
One came while cycling on shoulder of Interstate 25 about nine miles south of the WY/CO border. Cycling is legal and the shoulder is 10ft and there aren't too many other routes. I was heading southbound at about 7am on my recumbent bicycle after deciding to cut short a weekend tour (too much wind). As I approached mile marker 291, there was a truck that had ridden off the highway and rolled. The trucker was out of his truck, standing there as well as two Colorado state troopers with their vehicles parked on far side of shoulder.
I figured they had their hands full, so as I approached I moved to left edge of shoulder and just intended to cycle past. However as I got close, one of the state troopers yelled out "Get off the interstate!". So, I stopped. I politely explained that the CO bicycle map indicated this was legal, that back at mile marker 293 was a yellow diamond caution bicycles sign, and ahead at exit 288 was a small sign saying "bicycles must exit". I was a civil conversation and he must have thought I knew what I was talking about, so he also apologized and said "in seven years as a trooper I haven't seen anyone out here".
The other two police interactions I can think of, I had I was doing something wrong and was politely reminded. One was in Xinjiang, province China where I was cycling on the interstate equivalent highway despite No bicycles symbols (mostly because I didn't fully trust my send of directions on the small roads). I moved to the side road after being reminded.
The other example was one where I made a right turn after a stop sign - but also briefly paused looking left for other fellow cyclists. I wasn't paying attention as much as I should and actually pulled out not too far from a police car coming from the left. He turned on his lights, stopped me and said "what was that!". I was in the wrong, apologized and indicated I wasn't paying as much attention as I should. His verbal admonishment must have been enough because he let me go at that.