Police and aviation patches have become a hallmark of this country and the institutions that protect it. From the early days of golden stars in the wild west, that differentiated the good guys from the bad guys to the more modern stitched patches that represent the different ranks and academies.
Police patches have become a popular thing to to collect because the stitch work and variation between different locations has made a wide range of different types of patches. Coming in different shapes, patterns and colors each one varies quite a bit and makes for a great looking collection once you have over 25.
As I have mentioned before, most of the patches represent different eras and are nostalgic to people who lived through certain time periods. The police patches additionally are like passport stamps, and you can collect a patch from each of the 50 states, some of which are harder to locate and make for a great addition to the collection.
In a funny way its similar to those kids and their pokemon, except in this case we are talking about collectibles that have roots in history, and not some mythical creature represented by a card. Collectibles are collectibles however, but I prefer something with a bit more meaning.